User Guide Network In R

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Preface
Contents
1 Introducing Network Analysis in R
Part I Network Analysis Fundamentals
- 2 The Network Analysis `Five-Number Summary'
- 3 Network Data Management in R
Part II Visualization
Part III Description and Analysis
Part IV Modeling
References

Use R !

Douglas A. Luke

A User’s

Guide to

Network

Analysis in R

Use R!

Series Editors:

Robert Gentleman Kurt Hornik Giovanni Parmigiani

More information about this series at http://www.springer.com/series/6991

Use R!

Albert: Bayesian Computation with R (2nd ed. 2009)

Bivand/Pebesma/G´

omez-Rubio: Applied Spatial Data Analysis with R (2nd ed.

2013)

Cook/Swayne: Interactive and Dynamic Graphics for Data Analysis:

With R and GGobi

Hahne/Huber/Gentleman/Falcon: Bioconductor Case Studies

Paradis: Analysis of Phylogenetics and Evolution with R (2nd ed. 2012)

Pfaff: Analysis of Integrated and Cointegrated Time Series with R (2nd ed. 2008)

Sarkar: Lattice: Multivariate Data Visualization with R

Spector: Data Manipulation with R

Douglas A. Luke

A User’s Guide to Network

Analysis in R

123

Douglas A. Luke

Center for Public Health Systems Science

George Warren Brown School

of Social Work

Washington University

St.Louis,MO,USA

ISSN 2197-5736 ISSN 2197-5744 (electronic)

Use R!

ISBN 978-3-319-23882-1 ISBN 978-3-319-23883-8 (eBook)

DOI 10.1007/978-3-319-23883-8

Library of Congress Control Number: 2015955739

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To my most important social network—Sue,

Alina, and Andrew

Preface

In early 2000, Stephen Hawking said that “...the next century will be the century of

complexity.” If his prediction is true, the implication is that we will need new scien-

tiﬁc theories, data collection methods, and analytic techniques that are appropriate

for the study of complex systems and behavior. Network science is one such ap-

proach that views the world through a network lens, where physical and social sys-

tems are made up of heterogeneous actors who are connected to one another through

different types of relational ties. Network analysis is the set of analytic tools used

to study these types of systems. Over the past several decades network analysis has

become an increasingly important part of the analytic toolbox for social, health, and

physical scientists.

Until recently, network analysis required specialized software, both for network

data management and analyses. However, starting around 2000, network analytic

tools became available in the Rstatistical programming environment. This not only

made network analytic techniques more visible to the broader statistical community

but also provided the breadth and power of R’s data management, graphic visualiza-

tion, and general statistical modeling capabilities to the network analyst community.

As the title suggests, this book is a user’s guide to network analysis in R.Itpro-

vides a practical hands-on tour of the major network analytic tasks that can currently

be done in R. The book concentrates on four primary tasks that a network analyst

typically concerns herself with: network data management, network visualization,

network description, and network modeling. The book includes all the R code that is

used in the network analysis examples. It also comes with a set of network datasets

that are used throughout the book. (See Chap. 1for more details on the structure of

the book, as well as instructions on how to obtain the network data.) The book is

written for anybody who has an interest in doing network analysis in R. It can be

used as a secondary text in a network science or analysis class or can simply serve

as a reference for network techniques in R.

This book would not exist without the help, support, guidance, and mentoring

I have received over the last 30 years from my own personal and professional so-

cial networks. In the mid-1980s I took a graduate network analysis class from Stan

Wasserman at the University of Illinois in Champaign. I remember being excited

vii

viii Preface

about this new way to analyze data, but thought that I was not likely to ever use it in

my career. However, my colleagues in psychology and public health encouraged me

in my early work exploring how network analysis could answer important research

and evaluation questions. These include Julian Rappaport, Ed Seidman, Bruce Rap-

kin, Kurt Ribisl, Sharon Homan, Ross Brownson, and Matt Kreuter. Whether they

know it or not, I have been inspired and encouraged by an amazing group of net-

work and systems scientists, including Tom Valente, Steve Borgatti, Martina Morris,

Tom Snijders, Scott Leischow, Patty Mabry, Stephen Marcus, and Ross Hammond.

My best network ideas have come from my friends and colleagues at the Center for

Public Health Systems Science, particularly Bobbi Carothers, Amar Dhand, Chris

Robichaux, and Nancy Mueller. I am especially grateful to the students in my net-

work analysis classes and workshops over the years; they have not only improved

this book, but they have improved my thinking about network analysis. A very spe-

cial thank you to Jenine Harris. Jenine was my ﬁrst doctoral student, now I am

inspired by the rigor and elegance of her own work in network science. I would also

like to thank the Centers for Disease Control and Prevention, the National Insti-

tutes of Health, and the Missouri Foundation for Health for providing research and

evaluation support that allowed me to develop and reﬁne my approach to network

analysis. Finally, my deepest thanks go to my family. They gave me speciﬁc sug-

gestions about the content, provided me space and time to work hard on this book

(including a crucial Father’s Day gift), and cheered me on when I most needed it.

Thank you, Sue, Ali, and Andrew.

St. Louis, MO, USA Douglas A. Luke

July, 2015

Contents

1 Introducing Network Analysis in R ............................... 1

1.1 What Are Networks? ........................................ 1

1.2 What Is Network Analysis? .................................. 3

1.3 Five Good Reasons to Do Network Analysis in R ................ 4

1.3.1 Scope of R .......................................... 4

1.3.2 Free and Open Nature of R ............................ 5

1.3.3 Data and Project Management Capabilities of R .......... 5

1.3.4 Breadth of Network Packages in R ...................... 6

1.3.5 Strength of Network Modeling in R ..................... 6

1.4 Scope of Book and Resources ................................ 6

1.4.1 Scope .............................................. 6

1.4.2 Book Roadmap ...................................... 7

1.4.3 Resources .......................................... 8

Part I Network Analysis Fundamentals

2 The Network Analysis ‘Five-Number Summary’ ................... 11

2.1 Network Analysis in R: Where to Start ......................... 11

2.2 Preparation ................................................ 11

2.3 Simple Visualization ........................................ 12

2.4 Basic Description .......................................... 12

2.4.1 Size ............................................... 12

2.4.2 Density ............................................ 14

2.4.3 Components ........................................ 15

2.4.4 Diameter ........................................... 15

2.5 Clustering Coefﬁcient ....................................... 16

3 Network Data Management in R ................................. 17

3.1 Network Data Concepts ..................................... 17

3.1.1 Network Data Structures .............................. 17

3.1.2 Information Stored in Network Objects .................. 20

xContents

3.2 Creating and Managing Network Objects in R .................. 21

3.2.1 Creating a Network Object in statnet ................. 21

3.2.2 Managing Node and Tie Attributes...................... 24

3.2.3 Creating a Network Object in igraph .................. 28

3.2.4 Going Back and Forth Between statnet and igraph ... 30

3.3 Importing Network Data ..................................... 30

3.4 Common Network Data Tasks ................................ 32

3.4.1 Filtering Networks Based on Vertex or Edge Attribute

Values ............................................. 32

3.4.2 Transforming a Directed Network to a Non-directed

Network ............................................ 39

Part II Visualization

4 Basic Network Plotting and Layout .............................. 45

4.1 The Challenge of Network Visualization ....................... 45

4.2 The Aesthetics of Network Layouts ........................... 47

4.3 Basic Plotting Algorithms and Methods ........................ 49

4.3.1 Finer Control Over Network Layout .................... 50

4.3.2 Network Graph Layouts Using igraph ................. 52

5 Effective Network Graphic Design ............................... 55

5.1 Basic Principles ............................................ 55

5.2 Design Elements ........................................... 55

5.2.1 Node Color ......................................... 56

5.2.2 Node Shape ......................................... 60

5.2.3 Node Size .......................................... 62

5.2.4 Node Label ......................................... 66

5.2.5 Edge Width ......................................... 68

5.2.6 Edge Color ......................................... 69

5.2.7 Edge Type .......................................... 70

5.2.8 Legends ............................................ 71

6 Advanced Network Graphics .................................... 73

6.1 Interactive Network Graphics ................................ 73

6.1.1 Simple Interactive Networks in igraph ................ 74

6.1.2 Publishing Web-Based Interactive Network Diagrams ...... 74

6.1.3 Statnet Web: Interactive statnet with shiny .......... 77

6.2 Specialized Network Diagrams ............................... 77

6.2.1 Arc Diagrams ....................................... 78

6.2.2 Chord Diagrams ..................................... 79

6.2.3 Heatmaps for Network Data ........................... 82

6.3 Creating Network Diagrams with Other R Packages .............. 84

6.3.1 Network Diagrams with ggplot2 ..................... 84

Contents xi

Part III Description and Analysis

7 Actor Prominence .............................................. 91

7.1 Introduction ............................................... 91

7.2 Centrality: Prominence for Undirected Networks ................ 92

7.2.1 Three Common Measures of Centrality .................. 93

7.2.2 Centrality Measures in R .............................. 95

7.2.3 Centralization: Network Level Indices of Centrality ....... 96

7.2.4 Reporting Centrality .................................. 97

7.3 Cutpoints and Bridges .......................................101

8 Subgroups .....................................................105

8.1 Introduction ...............................................105

8.2 Social Cohesion ............................................106

8.2.1 Cliques .............................................107

8.2.2 k-Cores ............................................110

8.3 Community Detection .......................................115

8.3.1 Modularity ..........................................115

8.3.2 Community Detection Algorithms ......................118

9 Afﬁliation Networks ............................................125

9.1 Deﬁning Afﬁliation Networks ................................125

9.1.1 Afﬁliations as 2-Mode Networks .......................126

9.1.2 Bipartite Graphs .....................................126

9.2 Afﬁliation Network Basics ...................................127

9.2.1 Creating Afﬁliation Networks from Incidence Matrices ....127

9.2.2 Creating Afﬁliation Networks from Edge Lists ............129

9.2.3 Plotting Afﬁliation Networks ..........................130

9.2.4 Projections ..........................................131

9.3 Example: Hollywood Actors as an Afﬁliation Network ...........133

9.3.1 Analysis of Entire Hollywood Afﬁliation Network ........134

9.3.2 Analysis of the Actor and Movie Projections .............139

Part IV Modeling

10 Random Network Models .......................................147

10.1 The Role of Network Models .................................147

10.2 Models of Network Structure and Formation ....................148

10.2.1 Erd˝

os-R´

enyi Random Graph Model .....................148

10.2.2 Small-World Model ..................................151

10.2.3 Scale-Free Models ...................................154

10.3 Comparing Random Models to Empirical Networks ..............160

xii Contents

11 Statistical Network Models ......................................163

11.1 Introduction ...............................................163

11.2 Building Exponential Random Graph Models ...................165

11.2.1 Building a Null Model ................................167

11.2.2 Including Node Attributes .............................169

11.2.3 Including Dyadic Predictors ...........................171

11.2.4 Including Relational Terms (Network Predictors) .........175

11.2.5 Including Local Structural Predictors (Dyad Dependency) . . 177

11.3 Examining Exponential Random Graph Models .................179

11.3.1 Model Interpretation .................................179

11.3.2 Model Fit ...........................................180

11.3.3 Model Diagnostics ...................................183

11.3.4 Simulating Networks Based on Fit Model ................183

12 Dynamic Network Models .......................................189

12.1 Introduction ...............................................189

12.1.1 Dynamic Networks ...................................189

12.1.2 RSiena .............................................191

12.2 Data Preparation ...........................................192

12.3 Model Speciﬁcation and Estimation ...........................198

12.3.1 Speciﬁcation of Model Effects .........................198

12.3.2 Model Estimation ....................................203

12.4 Model Exploration .........................................203

12.4.1 Model Interpretation .................................203

12.4.2 Goodness-of-Fit .....................................209

12.4.3 Model Simulations ...................................212

13 Simulations ....................................................217

13.1 Simulations of Network Dynamics ............................217

13.1.1 Simulating Social Selection ............................218

13.1.2 Simulating Social Inﬂuence ............................228

References .........................................................235

Chapter 1

Introducing Network Analysis in R

Begin at the beginning, the King said, very gravely, and go on

till you come to the end: then stop. (Lewis Carroll, Alice in

Wonderland)

1.1 What Are Networks?

This book is a user’s guide for conducting network analysis in the Rstatistical

programming language. Networks are all around us. Humans naturally organize

themselves in networked systems. Our families and friends form personal social

networks around each of us. Neighborhoods and communities organize themselves

in networked coalitions to advocate for change. Businesses work with (and against)

each other in complex, interlocking networks of trade and ﬁnancial partnerships.

Public health is advanced through partnerships and coalitions of governmental and

NGO organizations (Luke and Harris 2007). Nations are connected to one another

through systems of migration, trade, and treaty obligations.

Moreover, non-human networks exist almost anywhere you look. Our genes and

proteins interact with one another through complex biological networks. The human

brain is now viewed as a complex network, or ‘connectome’ (Sporns 2012). Sim-

ilarly, human diseases and their underlying genetic roots are connected as a ‘dis-

easome’ (Barab´

asi 2007). Animal species interact in many complex ways, one of

which is a networked food-web that describes interactions in ‘who-eats-whom’ re-

lationships. Information itself is networked. Our legal system is built on an inter-

connecting network of prior legal decisions and precedents. Social and scientiﬁc

progress is driven by a diffusion of innovation process by which information is

disseminated across connected social systems, whether they are Iowa corn farmers

(Rogers 2003) or public health scientists (Harris and Luke 2009). It appears that one

of the ways the universe is organized is with networks.

So what is a network? Figures 1.1 and 1.2 present two examples of important

and interesting social networks. Figure 1.1 presents the contact network of the 19

9–11 hijackers, based on the work of Valdis Krebs (2002). Every social network

is made up of a set of actors (also called nodes) that are connected to one another

via some type of social relationship (also called a tie). In the ﬁgure, nodes are the

circles and the ties are the lines connecting some of the nodes. The network shows

D.A. Luke, A User’s Guide to Network Analysis in R, Use R!,

DOI 10.1007/978-3-319-23883-8 1

2 1 Introducing Network Analysis in R

us that the hijackers had some contact with one another before September 11th, but

the network is not very densely connected and there appears to be no prominent

network member who is connected to all or even most of the other hijackers.

AA11 (WTC North)

AA77 (Pentagon)

UA175 (WTC South)

UA93 (Pennsylvania)

Fig. 1.1 Network of 9–11 hijackers

The second example in Fig. 1.2 is from a very different sort of social network.

Here the nodes are members of the 2010 Netherlands FIFA World Cup team, who

went on to lose in the ﬁnal to Spain. The ties represent passes between the different

players during the World Cup matches. The arrows show the directional pattern of

the passes. We can see that the goalkeeper passed primarily to the defenders, and the

forwards received passes primarily from the midﬁelders (except for #6, who appears

to have a different passing pattern than the other two forwards).

These two examples may appear to have little in common. However, they both

share a fundamental characteristic common to all social networks. The social

patterns that are displayed in the network ﬁgures are not random. They reﬂect und-

erlying social processes that can be explored using network science theories and

methods. The terrorist network has no prominent leader and is not tightly inter-

connected because it makes the network harder to detect or disrupt. The pattern of

passing ties in the soccer network reﬂects the assigned positions of the players, the

rules of the game, and the strategies of the coach. The network analysis does not

‘know’ about any of those rules or strategies. Yet, network analysis can be used to

reveal these patterns that reﬂect the underlying rules and regularities.

1.2 What Is Network Analysis? 3

Defender

Forward

Goalkeeper

Midfielder

Fig. 1.2 Network of Netherlands 2010 World Cup soccer team

1.2 What Is Network Analysis?

Network science is a broad approach to research and scholarship that uses a rela-

tional lens to study and understand biological, physical, social, and informational

systems. The primary tool for network scientists is network analysis, which is a

set of methods that are used to (1) visualize networks, (2) describe speciﬁc charac-

teristics of overall network structure as well as details about the individual nodes,

ties, and subgroups within the networks, and (3) build mathematical and statistical

models of network structures and dynamics. Because the core question of network

science is about relationships, most of the methods used in network analysis are

quite distinct from the more traditional statistical tools used by social and health

scientists.

Network analysis as a distinct scientiﬁc enterprise with its own theories and

methods grew out of developments in many other disciplines, particularly graph

theory and topology in mathematics, the study of kinship systems in anthropology,

and social groups and process from sociology and psychology. Although network

analysis was not invented by one person at a speciﬁc place and time, the initial dev-

elopment of what we now recognize as modern network analysis can be traced back

to the work of Jacob Moreno in the 1930s. He deﬁned the study of social relations as

sociometry, and founded the journal Sociometry that would publish the early stud-

ies in this area. He also invented the sociogram, which was a visual way to display

4 1 Introducing Network Analysis in R

network structures. The ﬁrst published sociogram appeared in the New York Times

in 1933, and it was a network diagram of the friendship ties among a 4th grade class.

(These data are available as part of the network dataset package that accompanies

this book, see Sect. 1.4.3 below.)

The theories and methods of network analysis were developed throughout the rest

of the twentieth century, with important contributions from sociology, psychology,

political science, business, public health, and computer science. Network science

as an empirical practice was propelled by the development of a number of network

speciﬁc software tools and packages, including UCINet, STRUCTURE, Negopy,

and Pajek. The interest in network science has exploded in the last 20–30 years,

driven by at least three different factors. First, mathematicians, physicists, and other

researchers developed a number of inﬂuential theories of network structure and for-

mation that brought attention and energy to network science (see Chap. 10 for some

discussion of these theories). Second, advances in computational power and speed

allowed network methods to be applied to large and very large networks, such as

the internet, the population of the planet, or the human brain. Finally, advances in

statistical network theory allowed analysts for the ﬁrst time to move beyond sim-

ple network description to be able to build and test statistical models of network

structures and processes (see Chaps. 11 and 12).

1.3 Five Good Reasons to Do Network Analysis in R

As the title suggests, this book is designed as a general guide for how to do net-

work analysis in the R statistical language and environment. Why is R an ideal

platform for developing and conducting network analyses? There are at least ﬁve

good reasons.

1.3.1 Scope of R

The R statistical programming language and environment comprise a vast integrated

system of thousands of packages and functions that allow it to handle innumerable

data management, analysis, or visualization tasks. The R system includes a num-

ber of packages that are designed to accomplish speciﬁc network analytic tasks.

However, by performing these network tasks within the R environment, the analyst

can take advantage of any of the other capabilities of R. Most other network anal-

ysis programs (e.g., Pajek, UCINet, Gephi) are stand-alone packages, and thus do

not have the advantages of working within an integrated statistical programming

environment.

1.3 Five Good Reasons to Do Network Analysis in R 5

1.3.2 Free and Open Nature of R

One of the important reasons for R’s popularity and success is its free and open

nature. This is formally ensured via the GNU General Public License (GPL) that

R-code is released under. More informally, there is a vast R user and developer com-

munity which is continually working to enhance and improve R base code and the

thousands of R packages that can be freely accessed. The social network capabilities

of R described in this book have, in fact, been developed by the R user community.

This open nature of R facilitates faster (and arguably, cleaner and more powerful)

development and dissemination of new statistical and data analytic techniques, such

as these network analytic tools.

1.3.3 Data and Project Management Capabilities of R

Although there are many good network analysis programs available which can

handle a wide variety of network descriptive statistics and visualization tasks, no

other network package has the same power to handle often complex data and

project management tasks for larger-scale network analyses compared to R. First,

as suggested above, network analysis in R can take advantage of the powerful data

management, cleaning, import and export capabilities of base R. As described in

Chap. 3, network analysis often starts by importing and transforming data from

other sources into a form that can be analyzed by network tools. All network pack-

ages have some data management capabilities, but no other program can match R’s

breadth and depth.

Second, when conducting sophisticated scientiﬁc or commercial network anal-

yses, it is important to have the right project management tools to facilitate code

storage and retrieval, managing analysis outputs such as statistical results and infor-

mation graphics, and producing reports for internal and external audiences. Tradi-

tional statistical analysis platforms such as SAS and SPSS have these sorts of tools,

but most network programs do not. By pairing R up with an integrated development

environment (IDE) such as RStudio (http://rstudio.org/) and taking adv-

antage of packages such as knitr and shiny, the user has the ability to manage

any type of complex network project. In fact, the development and availability of

these tools has been one of the driving forces of the reproducible research move-

ment (Gentleman and Lang 2007), which emphasizes the importance of combining

data, code, results, and documentation in permanent and shareable forms. As one

example of the power of the reproducible research tools accessible in R is this book,

which was created entirely in RStudio.

6 1 Introducing Network Analysis in R

1.3.4 Breadth of Network Packages in R

The primary reason R is ideal for network analysis is the breadth of packages that

are currently available to manage network data and conduct network visualization,

network description, and network modeling. There are dozens of network-related

packages, and more are being created all the time. R network data can be managed

and stored in R native objects by the network and igraph packages, and the data

can be exchanged between formats with the intergraph package. Basic network

analysis and visualization can be handled with the sna package contained within

the much broader statnet suite of network packages, as well as within igraph.

More sophisticated network modeling can be handled by ergm and its associated

libraries, and dynamic actor-based network models are produced by RSiena. Free-

standing network analysis programs have many strengths (e.g., the visualization

capabilities of Gephi), but no single program matches the combined power of the

social network analysis packages contained in R.

1.3.5 Strength of Network Modeling in R

Finally, the particular network modeling strengths of R should be mentioned. R is

the only generally available software package that includes comprehensive facili-

ties to do stochastic network modeling (e.g., exponential random graph models),

dynamic actor-based network models that allow study of how networks change over

time, and other network simulation procedures.

1.4 Scope of Book and Resources

1.4.1 Scope

As the title suggests, the goal of this book is to provide a hands-on, practical guide to

doing network analysis in the Rstatistical programming environment. It is hands-on

in the sense that the book provides guidance primarily in the form of short network

analysis code snippets applied to realistic network data. The results of the analyses

follow immediately. All the code and data are available to the reader, so that it is

easy to replicate what is shown in the book, experiment with your own data or code

extensions, and thus facilitate learning.

The practical goal of the book is to demonstrate network analytic techniques

in Rthat will be useful for a wide variety of data analysis and research goals.

This includes data management, network visualization, computation of relevant net-

work descriptive statistics, and performing mathematical, statistical, and dynamic

1.4 Scope of Book and Resources 7

modeling of networks. The intended audiences include students, analysts and

researchers across a wide variety of disciplines, particularly the social, health,

business, and engineering domains.

It is also useful to state what this book is not designed to do. First, it does not

provide an in-depth treatment of network science theories or history. There are many

good books, papers, training courses, and online resources available that cover this

material. For good general overviews, the classic text by Wasserman and Faust

(1994) is still relevant, and John Scott provides a good, more current treatment

(2012). For more in-depth treatment of network science and statistical theory, see

Newman (2010) or Kolaczyk (2009). Finally, two edited volumes that have good

coverage of the recent history of network science as well as well-executed examples

of empirical network research are Newman et al. (2006) and Scott and Carrington

(2011).

Second, this book is not in any way an adequate introduction to Rprogramming

and statistical analysis. Although every attempt is made to make each code exam-

ple clear and succinct, a novice Ruser will ﬁnd some of the techniques and code

syntax hard to follow. In particular, understanding R’s capabilities for data manage-

ment, graphics, and the object-oriented approach to statistical modeling will be very

helpful for getting the most out of this user-guide.

Thus, the book is designed for the interested student, analyst, or researcher who is

familiar with Rand has some understanding of network science theories and meth-

ods. It could serve as a secondary text for a graduate level class in network analysis.

It also could be useful as a primer for an experienced Ranalyst who wants to incor-

porate network analysis into her programming and analytic toolbox.

1.4.2 Book Roadmap

The book is organized into four main sections, which correspond to the four

fundamental tasks that network analysts will spend most of their time on: data man-

agement, network visualization, network description, and network modeling. The

ﬁrst section has two chapters that cover both a simple introduction to basic net-

work techniques, then a more in-depth presentation of data management issues in

network analysis. The three chapters in the Visualization section cover basic net-

work graphics layout, network graphic design suggestions, and some discussion of

advanced graphics topics and techniques. The Description and Analysis section has

three chapters that cover the most widely used techniques for describing important

network characteristics, including actor prominence, network subgroups and com-

munities, and handling afﬁliation networks. The ﬁnal section, Modeling, includes

four chapters that present advanced techniques for mathematical modeling, statisti-

cal modeling, modeling of dynamic networks, and network simulations. Table 1.1

presents this roadmap.

8 1 Introducing Network Analysis in R

Chapter Packages Datasets

Introduction FIFA Nether, Krebs

5 number summary statnet, sna Moreno

Network data statnet, network, igraph DHHS, ICTS

Basic visualization statnet, sna Moreno, Bali

Graphic design statnet, sna, igraph Bali

Advanced graphics arcdiagram, circlize, visNetwork, networkD3 Simpsons, Bali

Prominence statnet, sna DHHS, Bali

Subgroups igraph DHHS, Moreno, Bali

Afﬁliation networks igraph hwd

Mathematical models igraph lhds

Stochastic models ergm TCnetworks

Dynamic models RSiena Coevolve

Simulations igraph

Table 1.1 User’s Guide roadmap

1.4.3 Resources

The most important resource for this user guide is a collection of network datasets

that have been curated and made available to the readers of this book. Over a dozen

network datasets are included in the form of an R package called UserNetR. These

datasets are used throughout the book to support the coding and analysis examples.

The network data included in the UserNetR package mostly come from published

network studies, while a few are created to help illustrate particular analytic options.

Table 1.1 lists the names of the datasets that are featured in each chapter.

The UserNetR package is maintained on GitHub, and must be downloaded and

installed to make the network data available. This can be done using the following

code. (The devtools package must also be installed if it is not on your system.)

library(devtools)

install_github("DougLuke/UserNetR")

Once this is done, the package must be loaded to make the various dataﬁles avail-

able. This can be done with the library() function, just like for any Rpackage.

This command will not always be explicitly shown throughout the book, so make

sure to load the package prior to executing any of the included Rcode.

library(UserNetR)

Finally, the documentation for the UserNetR package can be viewed through

the Rhelp system.

help(package='UserNetR')

Part I

Network Analysis Fundamentals

Chapter 2

The Network Analysis ‘Five-Number Summary’

There is nothing like looking, if you want to ﬁnd something. You

certainly usually ﬁnd something, if you look, but it is not always

quite the something you were after. (J.R.R. Tolkien –The

Hobbit)

2.1 Network Analysis in R: Where to Start

How should you start when you want to do a network analysis in R? The answer to

this question rests of course on the analytic questions you hope to answer, the state

of the network data that you have available, and the intended audience(s) for the

results of this work. The good news about performing network analysis in R is that,

as will be seen in subsequent chapters, R provides a multitude of available network

analysis options. However, it can be daunting to know exactly where to start.

In 1977, John Tukey introduced the ﬁve-number summary as a simple and quick

way to summarize the most important characteristics of a univariate distribution.

Networks are more complicated than single variables, but it is also possible to exp-

lore a set of important characteristics of a social network using a small number of

procedures in R.

In this chapter, we will focus on two initial steps that are almost always useful

for beginning a network analysis: simple visualization, and basic description using

a ‘ﬁve-number summary.’ This chapter also serves as a gentle introduction to basic

network analysis in R, and demonstrates how quickly this can be done.

2.2 Preparation

Similar to most types of statistical analysis using R, the ﬁrst steps are to load appro-

priate packages (installing them ﬁrst if necessary), and then making data available

for the analyses. The statnet suite of network analysis packages will be used

here for the analyses. The data used in this chapter (and throughout the rest of the

book) are from the UserNetR package that accompanies the book. The speciﬁc

dataset used here is called Moreno, and contains a friendship network of fourth

grade students ﬁrst collected by Jacob Moreno in the 1930s.

D.A. Luke, A User’s Guide to Network Analysis in R, Use R!,

DOI 10.1007/978-3-319-23883-8 2

12 2 The Network Analysis ‘Five-Number Summary’

library(statnet)

library(UserNetR)

data(Moreno)

2.3 Simple Visualization

The ﬁrst step in network analysis is often to just take a look at the network. Network

visualization is critical, but as Chaps. 4,5and 6indicate, effective network graph-

ics take careful planning and execution to produce. That being said, an informative

network plot can be produced with one simple function call. The only added com-

plexity here is that we are using information about the network members’ gender to

color code the nodes. The syntax details underlying this example will be covered in

greater depth in Chaps. 3,4and 5.

gender <- Moreno %v% "gender"

plot(Moreno, vertex.col = gender +2,vertex.cex =1.2)

The resulting plot makes it immediately clear how the friendship network is made

up of two fairly distinct subgroups, based on gender. A quickly produced network

graphic like this can often reveal the most important structural patterns contained in

the social network.

2.4 Basic Description

Tukey’s original ﬁve-number summary was intended to describe the most impor-

tant distributional characteristics of a variable, including its central tendency and

variability, using easy to produce statistical summaries. Similarly, using only a few

functions and lines of R code, we can produce a network ﬁve-number summary that

tells us how large the network is, how densely connected it is, whether the network

is made up of one or more distinct groups,howcompact it is, and how clustered are

the network members.

2.4.1 Size

The most basic characteristic of a network is its size. The size is simply the number

of members, usually called nodes, vertices or actors. The network.size() func-

tion is the easiest way to get this. The basic summary of a statnet network object

also provides this information, among other things. The Moreno network has 33

2.4 Basic Description 13

Fig. 2.1 Moreno sociogram

members, based on the network.size and summary calls. (Setting the print.adj

to false suppresses some detailed adjacency information that can take up a lot of

room.)

network.size(Moreno)

## [1] 33

summary(Moreno,print.adj=FALSE)

## Network attributes:

## vertices = 33

## directed = FALSE

## hyper = FALSE

## loops = FALSE

## multiple = FALSE

14 2 The Network Analysis ‘Five-Number Summary’

## bipartite = FALSE

## total edges = 46

## missing edges = 0

## non-missing edges = 46

## density = 0.0871

## Vertex attributes:

## gender:

## numeric valued attribute

## attribute summary:

## Min. 1st Qu. Median Mean 3rd Qu. Max.

## 1.00 1.00 2.00 1.52 2.00 2.00

## vertex.names:

## character valued attribute

## 33 valid vertex names

## No edge attributes

2.4.2 Density

Of all the basic characteristics of a social network, density is among the most imp-

ortant as well as being one of the easiest to understand. Density is the proportion

of observed ties (also called edges, arcs, or relations) in a network to the maximum

number of possible ties. Thus, density is a ratio that can range from 0 to 1. The

closer to 1 the density is, the more interconnected is the network.

Density is relatively easy to calculate, although the underlying equation differs

based on whether the network ties are directed or undirected. An undirected tie is

one with no direction. Collaboration would be a good example of an undirected tie;

if A collaborates with B, then by necessity B is also collaborating with A. Directed

ties, on the other hand, have direction. Money ﬂow is a good example of a directed

tie. Just because A gives money to B, does not necessarily mean that B reciprocates.

For a directed network, the maximum number of possible ties among kactors is

k∗(k−1), so the formula for density is:

k×(k−1),

where Lis the number of observed ties in the network. Density, as deﬁned here, does

not allow for ties between a particular node and itself (called a loop).

2.4 Basic Description 15

For an undirected network the maximum number of ties is k∗(k−1)/2 because

non-directed ties should only be counted once for every dyad (i.e., pair of nodes).

So, density for an undirected network becomes:

k×(k−1).

The information obtained in the previous section told us that the Moreno network

has 33 nodes and 46 non-directed edges. We could then use R to calculate that by

hand, but it is easier to simply use the gden() function.

den_hand <- 2*46/(33*32)

den_hand

## [1] 0.0871

gden(Moreno)

## [1] 0.0871

2.4.3 Components

A social network is sometimes split into various subgroups. Chapter 8will describe

how to use R to identify a wide variety of network groups and communities. How-

ever, a very basic type of subgroup in a network is a component. An informal def-

inition of a component is a subgroup in which all actors are connected, directly

or indirectly. The number of components in a network can be obtained with the

components function. (Note that the meaning of components is more compli-

cated for directed networks. See help(components) for more information.)

components(Moreno)

## [1] 2

2.4.4 Diameter

Although the overall size of a network may be interesting, a more useful character-

istic of the network is how compact it is, given its size and degree of interconnect-

edness. The diameter of a network is a useful measure of this compactness. A path

is the series of steps required to go from node A to node B in a network. The short-

est path is the shortest number of steps required. The diameter then for an entire

network is the longest of the shortest paths across all pairs of nodes. This is a mea-

sure of compactness or network efﬁciency in that the diameter reﬂects the ‘worst

16 2 The Network Analysis ‘Five-Number Summary’

case scenario’ for sending information (or any other resource) across a network.

Although social networks can be very large, they can still have small diameters

because of their density and clustering (see below).

The only complicating factor for examining the diameter of a network is that it

is undeﬁned for networks that contain more than one component. A typical app-

roach when there are multiple components is to examine the diameter of the largest

component in the network. For the Moreno network there are two components (see

Fig. 2.1). The smaller component only has two nodes. Therefore, we will use the

larger component that contains the other 31 connected students.

In the following code the largest component is extracted into a new matrix.

The geodesics (shortest paths) are then calculated for each pair of nodes using the

geodist() function. The maximum geodesic is then extracted, which is the dia-

meter for this component. A diameter of 11 suggests that this network is not very

compact. It takes 11 steps to connect the two nodes that are situated the furthest

apart in this friendship network.

lgc <- component.largest(Moreno,result="graph")

gd <- geodist(lgc)

max(gd$gdist)

## [1] 11

2.5 Clustering Coefﬁcient

One of the fundamental characteristics of social networks (compared to random

networks) is the presence of clustering, or the tendency to formed closed triangles.

The process of closure occurs in a social network when two people who share a

common friend also become friends themselves. This can be measured in a social

network by examining its transitivity. Transitivity is deﬁned as the proportion of

closed triangles (triads where all three ties are observed) to the total number of open

and closed triangles (triads where either two or all three ties are observed). Thus,

like density, transitivity is a ratio that can range from 0 to 1. Transitivity of a network

can be calculated using the gtrans() function. The transitivity for the 4th graders

is 0.29, suggesting a moderate level of clustering in the classroom network.

gtrans(Moreno,mode="graph")

## [1] 0.286

In the rest of this book, we will examine in more detail how the power of R can be

harnessed to explore and study the characteristics of social networks. The preceding

examples show that basic plots and statistics can be easily obtained. The meaning

of these statistics will always rest on the theories and hypotheses that the analyst

brings to the task, as well as history and experience doing network analysis with

other similar types of social networks.

Chapter 3

Network Data Management in R

Knowledge is of two kinds. We know a subject ourselves, or we

know where we can ﬁnd information upon it. (Samuel Johnson)

3.1 Network Data Concepts

A major advantage of using R for network analysis is the power and ﬂexibility of

the tools for accessing and manipulating the actual network data. One of the things

that I often tell my quantitative methods students is that they will typically spend the

majority of their time dealing with data management tasks and challenges. In fact,

the time spent analyzing and modeling data is dwarfed by the time spent getting

data ready for analyses. This is no different for network analysis. In fact, given the

specialized nature of network data, the data management tasks loom even larger. In

this chapter we cover three main topics. First, the general nature of network data

is explored and deﬁned. Second, we learn how network data objects can be created

and managed in R. Finally, a number of typical network data management tasks are

illustrated through a set of examples.

3.1.1 Network Data Structures

For many types of data analysis the data are stored in rectangular data structures,

where rows are used to depict cases or observations, and columns depict individual

variables. Spreadsheets use this type of data organization, as well as most statistics

packages such as SPSS. In R one of the fundamental data types is a ‘data frame,’

which uses this same rectangular format.

Networks, because of their need to depict more complicated relational structures,

require a different type of data storage. That is, in rectangular data structures the

fundamental piece of information is an attribute (column) of a case (row). In network

analysis, the fundamental piece of information is a relationship (tie) between two

members of a network.

Consider the following simple example of a directed network. The network

graphic itself depicts all of the information about the network. It is made up of

D.A. Luke, A User’s Guide to Network Analysis in R, Use R!,

DOI 10.1007/978-3-319-23883-8 3

18 3 Network Data Management in R

ﬁve nodes (named Athrough E), and there are a total of six directed ties. Because

these are directed ties, we can call them arcs (as compared to non-directed edges).

Although the network diagram is an efﬁcient way to communicate the network inf-

ormation to humans, computers need to use other methods to store, access, and

operate on the underlying network data.

Fig. 3.1 Simple directed network

3.1.1.1 Sociomatrices

Another way to depict the network data that is more useful for computer storage

is to arrange the information in a matrix. This type of matrix containing network

information is a sociomatrix. Table 3.1 contains the sociomatrix that corresponds

to Fig. 3.1. A sociomatrix is a square matrix where a 1 indicates a tie between two

nodes, and a 0 indicates no tie. So in Table 3.1 we see that there is a 1 in cell 1,2–this

indicates a tie going from node A to node B. The convention is that rows indicate

the starting node, and columns indicate the receiving node. A sociomatrix is also

sometimes called an adjacency matrix, because the 1s in the cells indicate which

nodes are adjacent to one another in the network.

If the network is non-directed (only edges instead of arcs), then the sociomatrix

would be symmetric around the diagonal. Here, however, cell 2,1 has a zero, indi-

cating that there is not an arc that goes from node B back to node A. For simple

networks, there are no self-loops, where a tie connects back to its own node. So,

diagonals are all zeros for simple networks.

3.1 Network Data Concepts 19

ABCDE

A01100

B00110

C01000

D00000

E00100

Table 3.1 Sociomatrix of the example directed network

3.1.1.2 Edge-Lists

Sociomatrices are elegant ways to depict networks, and they are a common way that

many network analysis programs store and manipulate network data. In particular,

many basic network algorithms are based on mathematical or statistical operations

on sociomatrices. For example, to ﬁnd geodesic distances between all pairs of nodes

in a network the underlying sociomatrices are multiplied together (Wasserman and

Faust 1994).

However, sociomatrices have one large disadvantage. As networks get larger,

sociomatrices become very sparse. That is, most of the matrix will be made up of

empty cells (cells with 0s). Table 3.2 shows the dramatic increase in both the size

and sparseness of a sociomatrix as the network size increases, keeping the average

degree constant at 3. This poses challenges for data storage, data manipulation, and

data display.

Nodes Avg. degree Edges Density Empty cells

10 3 15 0.33 70

100 3 150 0.03 9,700

1,000 3 1,500 0.00 997,000

Table 3.2 Demonstration of sparse sociomatrices

Fortunately, there is another way to depict network information that avoids this

problem of sociomatrices. Table 3.3 presents the edge list format for the example

network. As its name suggests, the edge list format depicts network information

by simply listing every tie in the network. Each row corresponds to a single tie,

that goes from the node listed in the ﬁrst column to the node listed in the second

column. Although the size of the sociomatrix and the edge list matrix are similar

for this small example (25 cells for the sociomatrix and 12 cells for the edge list

matrix), edge lists become much more efﬁcient for large networks. Referring back

to Table 3.2, for a network with 1,000 nodes, the sociomatrix would have 1,000,000

cells. The edge list for this network, with nodes having average degree of 3, would

only have 3,000 cells (1,500 edges between pairs of nodes).

20 3 Network Data Management in R

From To

Table 3.3 Edge list format for example directed network

3.1.2 Information Stored in Network Objects

Although basic matrices can be used to store some network information, R and other

statistics packages use more complex data structures to contain a wide variety of

network node, tie, metadata, and miscellaneous characteristics. In general, a network

data object can contain up to ﬁve types of information, as listed in Table 3.4.

Type Description Required?

Nodes List of nodes in network, along with node labels Required

Ties List of ties in the network Required

Node attributes Attributes of the nodes Optional

Tie attributes Attributes of the ties Optional

Metadata Other information about the entire network Depends

Table 3.4 Types of information contained in network data objects

First, a network data object must know which objects belong to the network, these

are generally known as nodes (in statnet they are called vertices). The second

required component in a network object is the list of ties that connect the nodes to

one another. Without these two types of information, the data object is not really a

network object. In addition to node and tie listings, network data objects will often

be able to store characteristics of those nodes and ties. For example, if the nodes

in the network are people, then basic information on those peoples such as gender

or income could be contained in the data object. Similarly, ties themselves may

have characteristics such as strength or valence (e.g., positive vs. negative). Finally,

network data objects may also contain metadata about the whole network or other

information that may be relevant or useful when accessing or analyzing the data.

For example, statnet stores global information about the network as metadata,

including whether the network is directed, whether loops are allowed, and whether

the network is bipartite.

3.2 Creating and Managing Network Objects in R 21

3.2 Creating and Managing Network Objects in R

Given R’s object-oriented design, it is not surprising that the main way that R

expects to access network data is through some type of a network data object. As

part of the statnet suite of packages, the network package deﬁnes a network

class that is an object structure designed to hold network data. Although statnet can

recognize relational data that are stored in basic matrices or data frames, much of

the power and ﬂexibility of R’s network analyses is unlocked when using network

data objects. For more detailed information about network objects in statnet,see

Butts (2008).

3.2.1 Creating a Network Object in

statnet

To create a network object, the identically-named network() function is called.

This function has a number of options, but the most common way to use it is to

feed relational data to it–typically an adjacency matrix or edge list. To see how this

works we will continue with the example directed network from Fig. 3.1. First, we

will create a network using an adjacency matrix.

netmat1 <- rbind(c(0,1,1,0,0),

c(0,0,1,1,0),

c(0,1,0,0,0),

c(0,0,0,0,0),

c(0,0,1,0,0))

rownames(netmat1) <- c("A","B","C","D","E")

colnames(netmat1) <- c("A","B","C","D","E")

net1 <- network(netmat1,matrix.type="adjacency")

class(net1)

## [1] "network"

summary(net1)

## Network attributes:

## vertices = 5

## directed = TRUE

## hyper = FALSE

## loops = FALSE

## multiple = FALSE

## bipartite = FALSE

## total edges = 6

## missing edges = 0

## non-missing edges = 6

## density = 0.3

22 3 Network Data Management in R

## Vertex attributes:

## vertex.names:

## character valued attribute

## 5 valid vertex names

## No edge attributes

## Network adjacency matrix:

## ABCDE

##A01100

##B00110

##C01000

##D00000

##E00100

The results of the class() and summary() calls show that we have success-

fully created a new network object. Also, this demonstrates that if the matrix has

identical row and column names, they will be used as the labels for the nodes. We

can also see that this is the same network as the earlier example by plotting it (Fig.

3.2).

gplot(net1, vertex.col =2,displaylabels =TRUE)

The same network can be created using an edge list format. This will often

be more convenient than adjacency matrices. Not only are edge lists smaller than

sociomatrices, but network data are often obtained naturally in this format. For

example, email communications can be analyzed as networks, where each email

corresponds to a tie from the email sender to the receiver. This leads easily to edge

list node pairs.

netmat2 <- rbind(c(1,2),

c(1,3),

c(2,3),

c(2,4),

c(3,2),

c(5,3))

net2 <- network(netmat2,matrix.type="edgelist")

network.vertex.names(net2) <- c("A","B","C","D","E")

summary(net2)

## Network attributes:

## vertices = 5

## directed = TRUE

## hyper = FALSE

## loops = FALSE

3.2 Creating and Managing Network Objects in R 23

Fig. 3.2 Plot of new network object

## multiple = FALSE

## bipartite = FALSE

## total edges = 6

## missing edges = 0

## non-missing edges = 6

## density = 0.3

## Vertex attributes:

## vertex.names:

## character valued attribute

## 5 valid vertex names

## No edge attributes

## Network adjacency matrix:

## ABCDE

##A01100

##B00110

##C01000

##D00000

##E00100

This produces the same network as before. Notice that the edgelist was provided

in the form of node ID numbers. To label the nodes properly, we used a special

vertex attribute constructor, network.vertex.names.

We have seen that to create network objects in R we can use a workﬂow that takes

data in a number of basic matrix formats and transforms them into the network class

24 3 Network Data Management in R

object. However, statnet also includes a number of tools that allow you to reverse

this workﬂow, by coercing network data into other matrix formats.

as.sociomatrix(net1)

## ABCDE

##A01100

##B00110

##C01000

##D00000

##E00100

class(as.sociomatrix(net1))

## [1] "matrix"

A more general coercion function is as.matrix(). It can be used to produce

a sociomatrix or an edgelist matrix.

all(as.matrix(net1) == as.sociomatrix(net1))

## [1] TRUE

as.matrix(net1,matrix.type ="edgelist")

## [,1] [,2]

## [1,] 1 2

## [2,] 3 2

## [3,] 1 3

## [4,] 2 3

## [5,] 5 3

## [6,] 2 4

## attr(,"n")

## [1] 5

## attr(,"vnames")

## [1] "A" "B" "C" "D" "E"

This ability to go back and forth between network objects and more fundamental

data structures such as sociomatrices and edgelist matrices gives the analyst great

power and ﬂexibility when managing network data. We will take advantage of these

tools later in this chapter as well as throughout the book.

3.2.2 Managing Node and Tie Attributes

One of the major advantages of using network objects when doing network anal-

ysis in R rather than using simpler matrix objects is the ability to store additional

3.2 Creating and Managing Network Objects in R 25

attribute information about the nodes and ties within the same network object. The

analyst typically knows much more about the members of a network than just the

simple list of nodes and ties. These node or tie characteristics can be used in network

visualization (see Chap. 5), network description, and network modeling (Chap. 11).

For both nodes and ties, statnet provides a set of functions that can be used to

create, delete, access, and list any attribute information of relevance. These functions

have a lot of capabilities, see help(attribute.methods) for more details.

3.2.2.1 Node Attributes

In the following example we use two different methods to set a pair of node att-

ributes (called vertex attributes by statnet). The ﬁrst example uses the more

formal method to assign gender codes to the nodes in net1. The second exam-

ple uses a shorthand method to assign a numeric vector as an attribute. In this case

we are storing the sum of the indegrees and outdegrees of each node as a new vertex

attribute.

set.vertex.attribute(net1, "gender",c("F","F","M",

"F","M"))

net1 %v% "alldeg" <- degree(net1)

list.vertex.attributes(net1)

## [1] "alldeg" "gender" "na"

## [4] "vertex.names"

summary(net1)

## Network attributes:

## vertices = 5

## directed = TRUE

## hyper = FALSE

## loops = FALSE

## multiple = FALSE

## bipartite = FALSE

## total edges = 6

## missing edges = 0

## non-missing edges = 6

## density = 0.3

## Vertex attributes:

## alldeg:

## numeric valued attribute

## attribute summary:

## Min. 1st Qu. Median Mean 3rd Qu. Max.

26 3 Network Data Management in R

## 1.0 1.0 2.0 2.4 4.0 4.0

## gender:

## character valued attribute

## attribute summary:

##FM

##32

## vertex.names:

## character valued attribute

## 5 valid vertex names

## No edge attributes

## Network adjacency matrix:

## ABCDE

##A01100

##B00110

##C01000

##D00000

##E00100

In this example, we see that information obtained outside of the network

(i.e., gender) or information obtained from the network itself (i.e., degree) can be

used as node attributes. Once node attributes have been set, they can be exam-

ined with the list.vertex.attributes command (note the plural). Also,

the summary of the network will provide some basic information about any stored

attributes.

To see the actual values stored in a vertex attribute, you can use the following

two equivalent methods.

get.vertex.attribute(net1, "gender")

## [1] "F" "F" "M" "F" "M"

net1 %v% "alldeg"

## [1] 2 4 4 1 1

3.2.2.2 Tie Attributes

Information about tie characteristics can also be stored and managed in the

network objects, using the similarly named set.edge.attributes and

get.edge.attributes functions. In the following example we create a new

edge attribute that contains a random number for each edge in the network, and then

access that information.

3.2 Creating and Managing Network Objects in R 27

list.edge.attributes(net1)

## [1] "na"

set.edge.attribute(net1,"rndval",

runif(network.size(net1),0,1))

list.edge.attributes(net1)

## [1] "na" "rndval"

summary(net1 %e% "rndval")

## Min. 1st Qu. Median Mean 3rd Qu. Max.

## 0.163 0.165 0.220 0.382 0.476 0.980

summary(get.edge.attribute(net1,"rndval"))

## Min. 1st Qu. Median Mean 3rd Qu. Max.

## 0.163 0.165 0.220 0.382 0.476 0.980

A more typical situation where you will want to create a new edge attribute is

when you are creating or working with valued networks. A valued network is one

where the network tie has some numeric value. For example, a resource exchange

network may include not just whether this is a ﬂow of money from one node to

another, but the actual amount of that money. In statnet, the actual values of

the valued ties are stored in an edge attribute. To see how this works, consider our

example network now as a friendship network, where the ﬁve network members

were asked to indicate how much they liked one another, on a scale of 0 (not at all)

to 3 (very much). The following example shows how we would proceed from the

raw valued sociomatrix to storing the values in an edge attribute called ‘like.’

netval1 <- rbind(c(0,2,3,0,0),

c(0,0,3,1,0),

c(0,1,0,0,0),

c(0,0,0,0,0),

c(0,0,2,0,0))

netval1 <- network(netval1,matrix.type="adjacency",

ignore.eval=FALSE,names.eval="like")

network.vertex.names(netval1) <- c("A","B","C","D","E")

list.edge.attributes(netval1)

## [1] "like" "na"

get.edge.attribute(netval1, "like")

## [1] 2 1 3 3 2 1

Thekeyherearetheignore.eval and names.eval options. These two

options, as set here, tell the network function to evaluate the actual values in the

28 3 Network Data Management in R

sociomatrix, and store those values in a new edge attribute called ‘like.’ Once values

are stored in an edge attribute, the original valued matrix can be restored using as

option of the as.sociomatrix coercion function.

as.sociomatrix(netval1)

## ABCDE

##A01100

##B00110

##C01000

##D00000

##E00100

as.sociomatrix(netval1,"like")

## ABCDE

##A02300

##B00310

##C01000

##D00000

##E00200

3.2.3 Creating a Network Object in

igraph

The other major R package that can be used to store and manipulate network data is

igraph, which is a comprehensive set of network data management and analytic

tools that have been implemented in R, Python, and C/C++. More information can

be obtained at igraph.org.

To start working with igraph, the package needs to be installed and loaded.

It contains a number of functions that have the same names as those found in the

statnet suite of packages, so it is a good idea to detach statnet before loading

igraph.

detach(package:statnet)

library(igraph)

For the most part, igraph can be used to store and access network, node, and

edge information in similar ways as the network package. In particular, igraph

network objects (called ‘graphs’) can be created from more basic sociomatrix or

edge list data structures.

inet1 <- graph.adjacency(netmat1)

class(inet1)

## [1] "igraph"

3.2 Creating and Managing Network Objects in R 29

summary(inet1)

## IGRAPH DN-- 5 6 --

## + attr: name (v/c)

str(inet1)

## IGRAPH DN-- 5 6 --

## + attr: name (v/c)

## + edges (vertex names):

## [1] A->B A->C B->C B->D C->B E->C

The summary information from an igraph graph object is slightly more cryptic

than from a statnet network. After the ‘IGRAPH’ tag is listed (indicating that

this is an igraph object), a series of codes are presented. In this case the ‘D’

indicates a directed graph, and the ‘N’ indicates that the vertices are named. Other

codes might appear that would designate whether the graph is weighted (i.e., valued)

or bipartite. After these codes the number of vertices (5) and edges (6) are then

displayed. See the help entry for summary.igraph for more details. The str()

function provides slightly more information, including the edge list.

Similarly, an igraph graph object can be created from an edge list.

inet2 <- graph.edgelist(netmat2)

summary(inet2)

## IGRAPH D--- 5 6 --

Node and tie attributes can be created, accessed, and transformed in similar ways

as within statnet. (In fact, management of node and tie attributes is somewhat

easier in igraph because of the underlying elegance of the accessor functions.) To

create and use node attributes, the V() vertex accessor function is used. Similarly,

to manage edge attributes, the E() edge accessor function is used. In this example

we use these functions to set names for the nodes, and to set edge values for the

observed ties.

V(inet2)$name <- c("A","B","C","D","E")

E(inet2)$val <- c(1:6)

summary(inet2)

## IGRAPH DN-- 5 6 --

## + attr: name (v/c), val (e/n)

str(inet2)

## IGRAPH DN-- 5 6 --

## + attr: name (v/c), val (e/n)

## + edges (vertex names):

## [1] A->B A->C B->C B->D C->B E->C

30 3 Network Data Management in R

3.2.4 Going Back and Forth Between

statnet

and

igraph

There will be times when you will want to use statnet network functions on

network data stored in an igraph graph object, and vice versa. To facilitate this,

the intergraph package can be used to transform network data objects between

the two formats. In the following example, we transform the net1 data into the

igraph format using the asIgraph function. If we wanted to go in the opposite

direction, we would use asNetwork.

library(intergraph)

class(net1)

## [1] "network"

net1igraph <- asIgraph(net1)

class(net1igraph)

## [1] "igraph"

str(net1igraph)

## IGRAPH D--- 5 6 --

## + attr: alldeg (v/n), gender (v/c), na

## | (v/l), vertex.names (v/c), na (e/l),

## | rndval (e/n)

## + edges:

## [1] 1->2 3->2 1->3 2->3 5->3 2->4

3.3 Importing Network Data

Importing raw data into R for subsequent network analyses is relatively straight-

forward, as long as the external data are in edge list, adjacency list, or sociomatrix

form (or can easily be transformed into such). This example creates an edge list that

corresponds to the same example network from Sect. 3.2.1 and then saves it as an

external CSV ﬁle. This ﬁle is then read in using read.csv and then turned into a

network data object.

detach("package:igraph",unload=TRUE)

library(statnet)

netmat3 <- rbind(c("A","B"),

c("A","C"),

c("B","C"),

3.3 Importing Network Data 31

c("B","D"),

c("C","B"),

c("E","C"))

net.df <- data.frame(netmat3)

net.df

## X1 X2

## 1 A B

## 2 A C

## 3 B C

## 4 B D

## 5 C B

## 6 E C

write.csv(net.df, file ="MyData.csv",

row.names =FALSE)

net.edge <- read.csv(file="MyData.csv")

net_import <- network(net.edge,

matrix.type="edgelist")

summary(net_import)

## Network attributes:

## vertices = 5

## directed = TRUE

## hyper = FALSE

## loops = FALSE

## multiple = FALSE

## bipartite = FALSE

## total edges = 6

## missing edges = 0

## non-missing edges = 6

## density = 0.3

## Vertex attributes:

## vertex.names:

## character valued attribute

## 5 valid vertex names

## No edge attributes

## Network adjacency matrix:

## ABCDE

##A01100

##B00110

##C01000

##D00000

32 3 Network Data Management in R

##E00100

gden(net_import)

## [1] 0.3

The network package in the statnet suite can read in external network data

that are in Pajek format (either Pajek .net or .paj ﬁles), using the read.paj()

function. The igraph package can also import Pajek ﬁles, as well as a few other

formats including GraphML and UCINet DL ﬁles.

3.4 Common Network Data Tasks

The preceding sections covered the basic information needed to create and manage

network data objects in R. However, the data managements tasks for network anal-

ysis do not end there. There are any number of network analytic challenges that will

require more sophisticated data management and transformation techniques. In the

rest of this chapter, two such examples are covered: preparing subsets of network

data for analysis by ﬁltering on node and edge characteristics, and turning directed

networks into non-directed networks.

3.4.1 Filtering Networks Based on Vertex or Edge Attribute Values

It is quite common to want to examine a subset of a network, either for quick visual-

ization or for further analyses. There are many ways to deﬁne or identify interesting

subnetworks in a larger network, and Chap. 8covers many of them. However, as

a basic data management task, you can ﬁlter a network based on values contained

either in edge attributes or vertex attributes. For both of these cases, you will delete

either the nodes or the edges, based on selection criteria that you set.

3.4.1.1 Filtering Based on Node Values

If a network object contains node characteristics, stored as vertex attributes, this

information can be used to select a new subnetwork for analysis. In our example

network we have the gender vertex attribute, so if you wanted to look at the sub-

network made up of females, you would use the following code (after switching

back from igraph to statnet).

3.4 Common Network Data Tasks 33

n1F <- get.inducedSubgraph(net1,

which(net1 %v% "gender" == "F"))

n1F[,]

## A B D

##A010

##B001

##D000

The get.inducedSubgraph() function returns a new network object that is

ﬁltered based on the vertex attribute criteria. This works because the %v% operator

returns a list of vertex ids.

gplot(n1F,displaylabels=TRUE)

The same process can work with numeric node characteristics. The following

code will plot the subset of the example network who all have degree greater than

or equal to 2. (But note that the nodes in the new subnetwork will of course not

have the same original degree values!) This works the same way but uses the %s%

operator, which is a shortcut for the get.inducedSubgraph function (Fig. 3.3).

Fig. 3.3 Female subnetwork

deg <- net1 %v% "alldeg"

n2 <- net1 %s% which(deg >1)

gplot(n2,displaylabels=TRUE)

34 3 Network Data Management in R

3.4.1.2 Removing Isolates

Another common ﬁltering task with networks is to examine the network after

removing all the isolates (i.e., nodes with degree of 0). We could use the

get.inducedSubgraph function from the previous section, but given that we

want to delete certain nodes we can take a more direct approach (Fig. 3.4).

Fig. 3.4 High degree subnetwork

For this short example, we will use the ICTS network dataset, which is available

as part of the UserNetR package that accompanies this book. The members of this

network are scientists, and they have a tie if they worked together on a scientiﬁc

grant submission. Using the isolates() function, we can see that this network

has a fair number of isolated nodes.

data(ICTS_G10)

gden(ICTS_G10)

## [1] 0.0112

length(isolates(ICTS_G10))

## [1] 96

The isolates() function returns a vector of vertex IDs. This can be fed to

the delete.vertices() function. However, unlike most R functions we have

seen, delete.vertices() does not return an object, but it directly operates on

the network that is passed to it. For that reason, it is safer to work on a copy of the

object.

n3 <- ICTS_G10

delete.vertices(n3,isolates(n3))

gden(n3)

3.4 Common Network Data Tasks 35

## [1] 0.0173

length(isolates(n3))

## [1] 0

3.4.1.3 Filtering Based on Edge Values

A social network often contains valued ties. For example, a resource exchange net-

work may list not only who exchanges money (or some other resource) with each

other, but the amount of money. Remember that in statnet information about ties

is stored in edge attributes (see Sect. 3.2.2). When a network has valued ties, it is not

unusual to want to examine the part of the network that only has certain values for

those ties. For example, you might want to visualize or analyze only those persons

in the resource exchange network who have given or received over a certain amount

of money. For this you will need to ﬁlter the network using tie values contained in

the appropriate edge attribute.

For this next example we will use a larger, more realistic social network. The

DHHS Collaboration Network (DHHS) contains network data from a study of the

relationships among 54 tobacco control experts working in 11 different agencies

in the Department of Health and Human Services in 2005. The main relationship

included in this dataset is collaboration – two members have a tie if they worked

together in the past year. This tie is valued to capture differences in the strength of

the collaboration. Speciﬁcally, the collaboration tie could take on one of four values:

(1) Shared information only; (2) Worked together informally; (3) Worked together

formally on a project; and (4) Worked together formally on multiple projects.

We can see that the raw network is relatively dense, and because of that the

network structure is somewhat hard to interpret when plotted (Fig. 3.5).

data(DHHS)

d<- DHHS

gden(d)

## [1] 0.312

op <- par(mar =rep(0,4))

gplot(d,gmode="graph",edge.lwd=d %e% 'collab',

edge.col="grey50",vertex.col="lightblue",

vertex.cex=1.0,vertex.sides=20)

par(op)

The graphic hard to interpret partly because of the high density, as well as having

some edge widths being thicker based on the value of the ‘collab’ attribute. We may

have a more interesting network graph (and one that is easier to interpret), if we

only examine the network ties for formal collaboration. That is, we can ﬁlter the

36 3 Network Data Management in R

Fig. 3.5 DHHS collaborations

original network and show only those ties where collaboration is coded 3 or higher.

To understand how edge ﬁltering works, it is important to remember how valued ties

are stored in a network object. The ties themselves are stored as a binary indicator

in the network object, while the values of those ties are stored in an edge attribute.

We can see how this works for the DHHS Collaboration network. First, we examine

the network ties for the ﬁrst six members of the network. Then we determine where

the collaboration values are stored, and then use that to view the tie values for the

same set of six actors.

as.sociomatrix(d)[1:6,1:6]

## ACF-1 ACF-2 AHRQ-1 AHRQ-2 AHRQ-3 AHRQ-4

## ACF-1 0 1 0 0 0 0

## ACF-2 1 0 0 0 0 0

## AHRQ-1 0 0 0 1 1 1

## AHRQ-2 0 0 1 0 1 1

## AHRQ-3 0 0 1 1 0 1

## AHRQ-4 0 0 1 1 1 0

list.edge.attributes(d)

## [1] "collab" "na"

as.sociomatrix(d,attrname="collab")[1:6,1:6]

3.4 Common Network Data Tasks 37

## ACF-1 ACF-2 AHRQ-1 AHRQ-2 AHRQ-3 AHRQ-4

## ACF-1 0 1 0 0 0 0

## ACF-2 1 0 0 0 0 0

## AHRQ-1 0 0 0 3 3 3

## AHRQ-2 0 0 3 0 3 2

## AHRQ-3 0 0 3 3 0 3

## AHRQ-4 0 0 3 2 3 0

The summary of the network object tells us that there are 447 ties in the DHHS

network. We can easily see the distribution of tie values.

table(d %e%"collab")

##1234

## 163 111 94 79

This indicates that of the 447 ties, 163 are informal sharing (1), 111 are informal

(2), 94 are formal on a single project (3), and the ﬁnal 79 ties are between DHHS

members who have worked together formally on multiple projects (4). Now we can

ﬁlter the edges to only include formal collaboration ties. This takes three steps. First,

a valued sociomatrix is created that contains the tie values stored in the ‘collab’ edge

attribute. Then we ﬁlter out the ties that we want to ignore. In this case the ties that

are coded 1 and 2 are replaced with 0s. Then, we create a new network based on the

ﬁltered sociomatrix. The key here is that a tie will be created anywhere a non-zero

value is found in d.val. Also, by using the ignore.eval and names.eval

options we store the retained edge values in an edge attribute called ‘collab.’

d.val <- as.sociomatrix(d,attrname="collab")

d.val[d.val <3]<- 0

d.filt <- as.network(d.val, directed=FALSE,

matrix.type="a",ignore.eval=FALSE,

names.eval="collab")

We can see that the new network has the same number of actors, but only 173 ties

(corresponding to the original numbers for the 3 and 4-levels of collab). Also, not

surprisingly, the density is now much lower.

summary(d.filt,print.adj=FALSE)

## Network attributes:

## vertices = 54

## directed = FALSE

## hyper = FALSE

## loops = FALSE

## multiple = FALSE

## bipartite = FALSE

38 3 Network Data Management in R

## total edges = 173

## missing edges = 0

## non-missing edges = 173

## density = 0.121

## Vertex attributes:

## vertex.names:

## character valued attribute

## 54 valid vertex names

## Edge attributes:

## collab:

## numeric valued attribute

## attribute summary:

## Min. 1st Qu. Median Mean 3rd Qu. Max.

## 3.00 3.00 3.00 3.46 4.00 4.00

gden(d.filt)

## [1] 0.121

Now when the network is plotted we can examine a smaller set of ties for impor-

tant structural information (Fig. 3.6).

op <- par(mar =rep(0,4))

gplot(d.filt,gmode="graph",displaylabels=TRUE,

vertex.col="lightblue",vertex.cex=1.3,

label.cex=0.4,label.pos=5,

displayisolates=FALSE)

par(op)

Note that the gplot() function itself has a limited ability to display only the

ties that exceed some lower threshold, using the thresh option. For example, this

command will display the same network as the previous code, without having to

go through the steps to create a new ﬁltered network (results not shown here). Note

that for this to work a valued sociomatrix has to be passed to gplot, not an actual

network object.

op <- par(mar =rep(0,4))

d.val <- as.sociomatrix(d,attrname="collab")

gplot(d.val,gmode="graph",thresh=2,

vertex.col="lightblue",vertex.cex=1.3,

label.cex=0.4,label.pos=5,

displayisolates=FALSE)

par(op)

3.4 Common Network Data Tasks 39

ACF-2

AHRQ-1

AHRQ-2

AHRQ-3

AHRQ-4

CDC-1

CDC-2

CDC-3

CDC-4

CDC-5

CDC-6

CDC-7

CDC-8

CDC-9

CDC-10

CDC-11

CDC-12

CMS-1

FDA-1

FDA-2

HRSA-1

HRSA-2

HRSA-3

IHS-1

IHS-2

NIH-1

NIH-2

NIH-3

NIH-4

NIH-5

NIH-6

NIH-7

NIH-8

NIH-9

NIH-10

NIH-11

NIH-12

NIH-13

NIH-14

NIH-16

OGC-1

OGC-2

OS-1

OS-2 OS-3

OS-4

OS-5

SAMHSA-3

Fig. 3.6 DHHS formal collaborations

3.4.2 Transforming a Directed Network to a Non-directed Network

It is often the case that even though the raw data in a network analysis is made up

of directed ties, the analyst wishes to consider the data as non-directed. This could

happen for several reasons. First, although the network relationship is non-directed,

the data collection procedures may result in directed ties. For example, in a survey

of collaboration among organizational representatives, even though collaboration is

non-directed (if agency A is collaborating with agency B, then B is also collabo-

rating with A), the raw data matrices are not likely to be perfectly symmetric. That

is, in self-report data, there may be error in the data or pairs or respondents may

not agree with each other on collaboration status. Respondent A may believe that

agency A collaborates with agency B, but respondent B may not believe the two

agencies are collaborating. In any case, you end up with a directed network that you

wish to ‘ﬁx’ by transforming it into a non-directed network.

40 3 Network Data Management in R

You may also wish to transform directed ties into non-directed ties for conceptual

reasons, and not simply to ﬁx data disagreements. For example, in studying trust

relationships you collect data on directed perceptions of trust. Here a tie between

A and B indicates that A trusts B. This is directed, in the sense that just because

A trusts B, that does not mean that B trusts A in return. However, you may wish

to analyze this in a non-directed sense, where an edge exists between two actors if

there is any trust relationship between the pair. So whether A trusts B, B trusts A, or

even if there is a reciprocal trusting relationship between A and B, then you would

treat A and B as having a trust relationship where you are ignoring the directionality

of the trust.

For either of these reasons, R makes it easy to transform a directed network

into a non-directed network. To do this you can use the symmetrize() function.

The name of the function should remind you that when network data are stored in

a sociomatrix, if the data are symmetric around the diagonal that indicates that the

ties are non-directed.

net1mat <- symmetrize(net1,rule="weak")

net1mat

## [,1] [,2] [,3] [,4] [,5]

## [1,] 0 1100

## [2,] 1 0110

## [3,] 1 1001

## [4,] 0 1000

## [5,] 0 0100

net1symm <- network(net1mat,matrix.type="adjacency")

network.vertex.names(net1symm) <- c("A","B","C","D","E")

summary(net1symm)

## Network attributes:

## vertices = 5

## directed = TRUE

## hyper = FALSE

## loops = FALSE

## multiple = FALSE

## bipartite = FALSE

## total edges = 10

## missing edges = 0

## non-missing edges = 10

## density = 0.5

## Vertex attributes:

## vertex.names:

3.4 Common Network Data Tasks 41

## character valued attribute

## 5 valid vertex names

## No edge attributes

## Network adjacency matrix:

## ABCDE

##A01100

##B10110

##C11001

##D01000

##E00100

The symmetrize procedure is relatively straightforward, except it returns a soc-

iomatrix (or, optionally, an edgelist). So we need to then turn it into a network object,

as we have done previously. The ‘rule’ option gives you four different choices on

how to symmetrize the ties. The ‘weak’ rule corresponds to a Boolean ‘OR’ condi-

tion where a tie is created between nodes i and j if there is a directed tie either from

i to j or from j to i. There is also a ‘strong’ rule, corresponding to a Boolean ‘AND’

where a tie is created between i and j only if there are directed ties from i to j and

j to i. This creates a symmetric network where the only ties preserved are the fully

reciprocated ties.

Part II

Visualization

Chapter 4

Basic Network Plotting and Layout

Above all else, show the data. (Edward R. Tufte, The Visual

Display of Quantitative Information)

4.1 The Challenge of Network Visualization

As suggested in Chap. 2, producing and examining a network plot is often one of

the ﬁrst steps in network analysis. The overall purpose of a network graphic (as with

any information graphic) is to highlight the important information contained in the

underlying data. However, there are innumerable ways to visually layout network

nodes and ties in two-dimensional space, as well as using graphical elements (e.g.,

node size, line color, ﬁgure legend, etc.) to communicate the story in the network

data. In the next three chapters we go over basic principles of effective network

graph design, and how to produce effective network visualizations in R.

An effective network graphic will convey the important information in a social

network, such as the overall structure, location of important actors in the network,

presence of distinctive subgroups, etc. At the same time, the graphic should do its

best to minimize irrelevant information. For example, tie length in a network graphic

is arbitrary in the sense that the length of a tie is not meaningful. An effective net-

work ﬁgure will be designed and laid out in a way that minimizes the chance that a

viewer will misinterpret the meaning of tie lengths.

The purpose of this chapter is to introduce basic plotting techniques for networks

in R, and discuss the various options for specifying the layout of the network on

the screen or page. The following example shows how interpretation of a network

graphic can be impeded or enhanced by its basic layout.

data(Moreno)

op <- par(mar =rep(0,4),mfrow=c(1,2))

plot(Moreno,mode="circle",vertex.cex=1.5)

plot(Moreno,mode="fruchtermanreingold",vertex.cex=1.5)

par(op)

At ﬁrst glance it may appear that the ﬁgures are showing two quite different net-

works. In fact, they are two different visual representations of the same underlying

D.A. Luke, A User’s Guide to Network Analysis in R, Use R!,

DOI 10.1007/978-3-319-23883-8 4

46 4 Basic Network Plotting and Layout

Fig. 4.1 Same network, different layouts

social network, in this case the friendship ties among a 4th grade class. Despite rep-

resenting the same network data, the righthand ﬁgure is easier for us to interpret. In

particular, it is much easier to see that the network is made up of two separate com-

ponents, and that the large component has two fairly distinct cohesive subgroups.

That is, the important structural characteristics of the network are easier to deter-

mine with the second layout compared to the ﬁrst.

Although it is possible to lay out a network in 3D-space, the vast majority of

network visualizations are two-dimensional. Nodes are represented by shapes, typ-

ically circles, and ties are represented by straight or sometimes curved lines. The

lines themselves can be tricky to interpret for somebody new to network visualiza-

tion. In particular, the length of the line has no real meaning. Consider the following

two graphs, which display the same simple network (Fig. 4.2). At a quick glance it

might appear that node D is further away from B and C in the second graph. But the

ties simply indicate which nodes are adjacent to one another, so the length of each

line does not communicate any substantive information.

D A

Fig. 4.2 Line length is arbitrary

However, as the Moreno 4th grade friendship network example illustrated

(Fig. 4.1), despite the arbitrary nature of some of the layout elements, the way a

network is depicted in a graphic can enhance or obscure other important structural

information.

4.2 The Aesthetics of Network Layouts 47

This is the fundamental challenge of network visualization: to reveal important

structural characteristics of the network without distortion or as Edward Tufte stated,

The minimum we should hope for with any display technology is that it should do

no harm (Tufte 1990).

4.2 The Aesthetics of Network Layouts

Although there are not in fact an inﬁnite number of ways to display a network on

a screen, the number of possibilities might as well be. (For example, consider a

moderate sized network of 50 nodes, and a display grid of 10 by 10. In actuality,

the display grid would be much larger than this. The ﬁrst node in the network could

be placed in any one of 100 positions, the 2nd node in 99 positions, and so on.

In this example, there are 3.1×1093 different possible network layouts.) Most of

the possibilities will produce ugly or confusing layouts, therefore there must be

some way to pick a layout that has a better than average chance of being visually

acceptable.

Fortunately, network and visualization scientists have studied what makes network

graph layouts easier to understand and interpret. What has emerged from this line of

work is a set of aesthetic principles that can be used to more effectively display net-

works. Network graphics are easier to understand if they follow as much as possible

the following ﬁve guidelines:

• Minimize edge crossings.

• Maximize the symmetry of the layout of nodes.

• Minimize the variability of the edge lengths.

• Maximize the angle between edges when they cross or join nodes.

• Minimize the total space used for the network display.

A large number of approaches have been developed for automatic layout of

network graphics. One general class of algorithms, called force-directed, has proven

to be a ﬂexible and powerful approach to automatic network layouts. These algo-

rithms work iteratively to minimize the total energy in a network, where the en-

ergy can be deﬁned in a number of ways. A popular approach is to have connected

nodes have a spring-like attractive force, while simultaneously assigning repulsive

forces to all pairs of nodes. The springs in this algorithm act to pull connected nodes

closer to one another, while the repulsive forces push unconnected nodes away from

each other. The resulting network system will move around and oscillate for a while

before settling into a steady state that tends to minimize the energy in the network

system. This describes how the algorithm works, but the remarkable feature is that

the resulting network graph tends to produce displays that are aesthetically pleasing,

in the sense described above (Fruchterman and Reingold 1991).

To see the positive results of using one of these algorithms, consider the compar-

ison in Fig. 4.3. On the left-hand side, the Moreno network is displayed randomly.

On the right-hand side we are using the Fruchterman-Reingold algorithm for the net-

work display. Fruchterman and Reingold introduced one of the ﬁrst force-directed

48 4 Basic Network Plotting and Layout

network display algorithms, and it is still very widely used. In fact, it is the default

algorithm used by the statnet network plotting functions. On the right-hand side

the nodes are displayed more symmetrically, there are relatively fewer edge cross-

ings, and the tie lengths are more uniform. All of this makes it easier to interpret the

structural information contained in the network.

op <- par(mar =c(0,0,4,0),mfrow=c(1,2))

gplot(Moreno,gmode="graph",mode="random",

vertex.cex=1.5,main="Random layout")

gplot(Moreno,gmode="graph",mode="fruchtermanreingold",

vertex.cex=1.5,main="Fruchterman-Reingold")

par(op)

Random layout Fruchterman-Reingold

Fig. 4.3 Moreno network-random vs. Fruchterman-Reingold layouts

As stated above, a force-directed algorithm works by iteratively adjusting the

overall network layout until some measure of overall network energy is minimized.

The details of this are usually not of interest, but to see how this works in practice

consider Fig. 4.4, which displays the Bali terrorist network. Starting from a circle

layout, it shows how the Fruchterman-Reingold layout algorithm works through

successive iterations, from 0 (the starting circle) to 50.

The Fruchterman-Reingold algorithm, along with other force-directed approaches,

are iterative and non-deterministic. That means that each time you run the plotting

algorithm you will not get the exact same layout. However, you will get a layout

that tends to be symmetrical, minimize edge crossings, etc.

4.3 Basic Plotting Algorithms and Methods 49

Iteration = 0 Iteration = 1 Iteration = 5

Iteration = 10 Iteration = 25 Iteration = 50

Fig. 4.4 Iterative Fruchterman-Reingold algorithm

4.3 Basic Plotting Algorithms and Methods

Network visualization in statnet is handled by two closely related functions,

plot and gplot. The latter has more layout options, so it may be more gener-

ally useful. To use a different layout algorithm, it is as simple as specifying the

appropriate layout option. Figure 4.5 shows six of the layout options for the gplot

function.

op <- par(mar=c(0,0,4,0),mfrow=c(2,3))

gplot(Bali,gmode="graph",edge.col="grey75",

vertex.cex=1.5,mode='circle',main="circle")

gplot(Bali,gmode="graph",edge.col="grey75",

vertex.cex=1.5,mode='eigen',main="eigen")

gplot(Bali,gmode="graph",edge.col="grey75",

vertex.cex=1.5,mode='random',main="random")

gplot(Bali,gmode="graph",edge.col="grey75",

vertex.cex=1.5,mode='spring',main="spring")

gplot(Bali,gmode="graph",edge.col="grey75",

vertex.cex=1.5,mode='fruchtermanreingold',

50 4 Basic Network Plotting and Layout

main='fruchtermanreingold')

gplot(Bali,gmode="graph",edge.col="grey75",

vertex.cex=1.5,mode='kamadakawai',

main='kamadakawai')

par(op)

circle eigen random

spring fruchtermanreingold kamadakawai

Fig. 4.5 Network layout options

4.3.1 Finer Control Over Network Layout

The layout options provided in statnet (and igraph, see below) work algo-

rithmically or heuristically, usually with some randomness. So, even with the same

layout option, a different graphic layout will be produced each time the network is

plotted. Fortunately, R provides a way to have exact control over the layout coor-

dinates. This allows for exact positioning, or saving the layout coordinates after a

particular network is plotted.

The coord option in the plot function is used for this. This option expects a

matrix with two columns. Each row corresponds to one node, the ﬁrst column gives

the X coordinate, and the second column gives the Y coordinate. Also, the results

4.3 Basic Plotting Algorithms and Methods 51

of a plotting function can be saved to an object, which will contain the coordinates

of the produced plot. This is demonstrated in the next example. Here, we produce

an initial plot of the Bali network, saving the coordinates. We then stretch out the

layout of the graph by multiplying the Y coordinates by a constant. Both plots are

shown in the ﬁgure, along with axes to make it easier to see how the coordinates

have changed (Fig. 4.6). There are many other ways to use speciﬁc coordinates, but

the main use is to preserve a particular layout for future production and examination.

mycoords1 <- gplot(Bali,gmode="graph",

vertex.cex=1.5)

mycoords2 <- mycoords1

mycoords2[,2]<- mycoords1[,2]*1.5

mycoords1

## x y

## [1,] -6.299 11.84

## [2,] -3.887 13.80

## [3,] -8.355 9.89

## [4,] -4.672 10.28

## [5,] -8.537 11.65

## [6,] -5.932 12.63

## [7,] -2.420 12.96

## [8,] -7.694 10.09

## [9,] -8.334 10.86

## [10,] -0.935 8.08

## [11,] -3.015 6.98

## [12,] -1.863 7.10

## [13,] -1.094 9.30

## [14,] -2.061 8.51

## [15,] -7.715 12.83

## [16,] -10.453 11.25

## [17,] -8.357 12.43

mycoords2

## x y

## [1,] -6.299 17.8

## [2,] -3.887 20.7

## [3,] -8.355 14.8

## [4,] -4.672 15.4

## [5,] -8.537 17.5

## [6,] -5.932 18.9

## [7,] -2.420 19.4

## [8,] -7.694 15.1

## [9,] -8.334 16.3

## [10,] -0.935 12.1

52 4 Basic Network Plotting and Layout

## [11,] -3.015 10.5

## [12,] -1.863 10.7

## [13,] -1.094 14.0

## [14,] -2.061 12.8

## [15,] -7.715 19.2

## [16,] -10.453 16.9

## [17,] -8.357 18.6

op <- par(mar=c(4,3,4,3),mfrow=c(1,2))

gplot(Bali,gmode="graph",coord=mycoords1,

vertex.cex=1.5,suppress.axes =FALSE,

ylim=c(min(mycoords2[,2])-1,max(mycoords2[,2])+1),

main="Original coordinates")

gplot(Bali,gmode="graph",coord=mycoords2,

vertex.cex=1.5,suppress.axes =FALSE,

ylim=c(min(mycoords2[,2])-1,max(mycoords2[,2])+1),

main="Modified coordinates")

par(op)

4.3.2 Network Graph Layouts Using

igraph

The igraph package provides the user with a similar set of options for controlling

the layouts of network graphics. The layout option is used to specify an existing

layout function or refer to a set of vertex coordinates. See ?igraph.plotting

for more information on plotting and layout options in igraph (Fig. 4.7).

detach(package:statnet)

library(igraph)

library(intergraph)

iBali <- asIgraph(Bali)

op <- par(mar=c(0,0,3,0),mfrow=c(1,3))

plot(iBali,layout=layout_in_circle,

main="Circle")

plot(iBali,layout=layout_randomly,

main="Random")

plot(iBali,layout=layout_with_kk,

main="Kamada-Kawai")

par(op)

4.3 Basic Plotting Algorithms and Methods 53

Original coordinates

-12 -8-6-4-20

5 1015202530

-12 -8-6-4-20

5 1015202530

Modified coordinates

Fig. 4.6 Network layouts with modiﬁed coordinates

Circle

13 14 15

Random

13 14

Kamada-Kawai

Fig. 4.7 Network layout options with igraph

Chapter 5

Effective Network Graphic Design

As with any graphic, networks are used in order to discover

pertinent groups or to inform others of the groups and structures

discovered. It is a good means of displaying structures.

However, it ceases to be a means of discovery when the elements

are numerous. The ﬁgure rapidly becomes complex, illegible

and untransformable. (Jacques Bertin)

5.1 Basic Principles

Achieving effective network graphic design is not that different from any other type

of information graphic. As Edward Tufte pointed out in his seminal The Visual Dis-

play of Quantitative Information, “Graphical excellence is that which gives to the

viewer the greatest number of ideas in the shortest time with the least ink in the

smallest space.” Network graphics actually start out with an important advantage in

that they typically have a high information/ink ratio.

The goal for any network graphic design should be to produce a ﬁgure that reveals

the important or interesting information that is contained in the network data. To

do this, the analyst must make decisions about every graphical element that can

appear in the ﬁgure. R, and the plotting functions contained in the statnet and

igraph packages, give the analyst almost complete programmatic control over the

appearance of the network graphic. The purpose of this chapter is to walk through

many of the most useful design elements in network graphics, and discuss how to

use them and why they should be used in certain ways.

5.2 Design Elements

Like any other type of information graphic, network visualizations are made up

of a large number of distinct visual elements. These individual elements include

things that are distinctive to network graphics, such as nodes and ties, as well as

other elements common to most graphics, such as titles, legends, etc. The plotting

functions in statnet and igraph provide a great deal of programmatic control

to the user.

Although a simple call to a plotting function is enough to produce a default net-

work graphic, it is almost always the case that you will need to take time to set

appropriate function options and develop some additional Rcode to produce an

D.A. Luke, A User’s Guide to Network Analysis in R, Use R!,

DOI 10.1007/978-3-319-23883-8 5

56 5 Effective Network Graphic Design

effective graphic. Some design decisions will be made on aesthetic grounds, while

many others will be based on the most important pattern or story that you wish to

convey with the graphic and that is supported by the underlying network data.

The following sections present a quick tour of the most commonly used indi-

vidual graphing elements for network visualizations. They will each be covered on

their own in turn.

5.2.1 Node Color

By default, statnet produces a network graphic with red circles as nodes. To

designate a different color, the vertex.col option is used in the gplot function.

(The gmode option is also used here to tell statnet not to handle Bali as a

directed graph.) So, for example, it is a simple matter to produce a plot with

attractive light blue nodes (Fig. 5.1).

data(Bali)

gplot(Bali,vertex.col="slateblue2",gmode="graph")

In general, all of the basic color-handling options of Rare available for plotting

networks. This opens up a lot of power and ﬂexibility for graphic design, but to use

color effectively will require some homework. In particular, it will be useful to read

more in-depth treatments of color use in R(e.g., Murrell 2005).

As the above example suggests, a color can be designated by its color name.

To see all of the 657 possible color names recognized by R,usethecolors()

command. In addition to specifying the name, colors can be selected using Red-

Green-Blue (RGB) triplets of intensities. Alternatively, the RGB speciﬁcation can

also be provided using a hexadecimal string of the form ‘#RRGGBB’, where each

of the RR, GG, and BB parts of the string is a hexadecimal number that provides

the red, green or blue intensity ranging from 00 to FF.

The following code will produce the same network graphic with the same light

blue nodes (ﬁgure not shown), showing how you can obtain colors using the rgb

and hexadecimal approaches. To get the appropriate rgb values for a particular color

name, you can use the col2rgb() function. The hexadecimal codes were obtained

at http://www.javascripter.net/faq/rgbtohex.htm.

col2rgb('slateblue2')

gplot(Bali,vertex.col=rgb(122,103,238,

maxColorValue=255),gmode="graph")

gplot(Bali,vertex.col="#7A67EE",gmode="graph")

One less common color feature in Rcan come in handy for network diagrams,

especially with large networks where the nodes overlap in the graphic. Normally,

colors are fully opaque, so overlapping nodes in a graphic will lead to large color

‘blobs’ where it is hard to distinguish the nodes. However, it is possible to make

5.2 Design Elements 57

Fig. 5.1 Bali network with light blue nodes

use of partially transparent colors, using a built-in alpha transparency channel. The

rgb() function can be used to specify the amount of transparency, from 0 (fully

transparent) to 1 (fully opaque). See ?rgb for more details.

Figure 5.2 shows the difference when using the alpha transparency color channel.

Both graphics show the same random network of 300 nodes. (The layouts are differ-

ent because each is using the Fruchterman-Reingold force-directed algorithm.) The

ﬁgure on the left is using a fully opaque dark blue color. The ﬁgure on the right is

still using dark blue, but with an alpha transparency channel of approximately 30 %.

The overlapping nodes are much easier to see when transparent colors are used.

Note that some graphics devices in Rmay not support transparent colors.

ndum <- rgraph(300,tprob=0.025,mode="graph")

op <- par(mar =c(0,0,2,0),mfrow=c(1,2))

gplot(ndum,gmode="graph",vertex.cex=2,

vertex.col=rgb(0,0,139,maxColorValue=255),

edge.col="grey80",edge.lwd=0.5,

main="Fully opaque")

gplot(ndum,gmode="graph",vertex.cex=2,

vertex.col=rgb(0,0,139,alpha=80,

maxColorValue=255),

edge.col="grey80",edge.lwd=0.5,

main="Partly transparent")

par(op)

In these previous examples, every node has the same color. A more important

use of color is to communicate some characteristic of the node or network by having

different nodes have different colors. Speciﬁcally, information stored in a categorical

node attribute can often be communicated through judicious node color choices.

58 5 Effective Network Graphic Design

Fully opaque Partly transparent

Fig. 5.2 Alpha transparency channel example

For example, the Bali terrorist network has the role vertex attribute which

stores the categorical description of the role that each member played in the net-

work. CT means a member of the command team, BM is a bomb maker, etc. (See

?Bali for more information.) Node colors can be used to effectively distinguish

the network member roles. Since this information is already stored in a vertex att-

ribute, statnet can use this to automatically pick node colors. (This is only true

for plot(), not gplot().)

rolelab <- get.vertex.attribute(Bali,"role")

op <- par(mar=c(0,0,0,0))

plot(Bali,usearrows=FALSE,vertex.cex=1.5,label=rolelab,

displaylabels=T,vertex.col="role")

par(op)

Figure 5.3 displays the Bali network with nodes colored according to the role

each member played in the network. (The node labels are also printed out to

facilitate interpretation. Node labeling will be discussed in Sect. 5.2.4.) The net-

work is much more interpretable by using color coding in this way. For example,

we can more easily understand the subgroup structure by noting that the greater

density between the members of Team Lima (TL, cyan), as well as the bombmakers

(BM, black).

However, by simply using the name of an existing vertex attribute, statnet

picks node colors from the existing default color palette in R. Viewing this palette,

we can see that “BM” is assigned to the color black because “BM” comes ﬁrst

alphabetically in the role attribute string, and black is the ﬁrst entry in the color

palette. “CT” comes second, so it is assigned red which is the 2nd entry in the

palette, an so on.

5.2 Design Elements 59

palette()

## [1] "black" "red" "green3" "blue"

## [5] "cyan" "magenta" "yellow" "gray"

Using the default palette has a number of disadvantages. First, it is limited to

eight colors. (Rwill cycle through the set of eight colors if there are more than eight

types of nodes to color.) Second, the default palette starts with black which is often

not a good color choice to include with other colors in a network graphic. More

generally, the default colors do not represent an aesthetically pleasing or useful set

of colors for displaying categorical classiﬁcations.

A more ﬂexible, and usually aesthetically more satisfying approach is to set up

your own color palette and then index into it for color selection. The RColor

Brewer package provides a number of predesigned color palettes that are very

useful when using color to distinguish between a relatively small set of categories.

For more information, see ?RColorBrewer. More details on the ColorBrewer

Fig. 5.3 Bali network colored by role

60 5 Effective Network Graphic Design

system are available at http://www.colorbrewer.org. (There are many

other color and palette picking options available, for example see the interactive

palette chooser at http://paletton.com.)

library(RColorBrewer)

display.brewer.pal(5,"Dark2")

In the following code, a user-deﬁned palette is created by selecting ﬁve colors

from a larger palette called Dark2 provided by RColorBrewer. Once the palette

has been deﬁned, it can be used in the network plotting call. This approach produces

more pleasing sets of colors, and is much more ﬂexible than relying on the default

color palette (Fig. 5.4).

Dark2 (qualitative)

Fig. 5.4 A set of ﬁve colors chosen from an RColorBrewer palette

Note that we convert the role vertex attribute character vector to a factor so

that the indexing will work. This means that if you have a numeric vector stored as

a vertex attribute that you do not have to turn it into a factor. In other words, the

indexing works with factors or numeric vectors, but not character vectors (Fig. 5.5).

my_pal <- brewer.pal(5,"Dark2")

rolecat <- as.factor(get.vertex.attribute(Bali,"role"))

plot(Bali,vertex.cex=1.5,label=rolelab,

displaylabels=T,vertex.col=my_pal[rolecat])

5.2.2 Node Shape

In addition to using color to distinguish between different types of nodes, statnet

can be directed to use different shapes for the nodes. This is mainly useful when

5.2 Design Elements 61

Fig. 5.5 Bali network with better colors

there is a small number of node types. It is also particularly useful for situations

where you will not be able to use color to distinguish nodes (or help viewers who

may be color-blind).

Unfortunately, statnet has only a limited ability to distinguish nodes by

shapes, by designating the number of sides used to plot the node polygon (normally,

the number of sides is 50, which produces a circle). If the number of sides is 3 you

get a triangle, 4 a square, and so on. This is only useful for a very small number of

node types (Fig. 5.6).

If you have a particular need to use node shapes in a network graphic, igraph is

much more ﬂexible in this regard. See Sect. 9.2.3 for an igraph plotting example

with different node shapes.

op <- par(mar=c(0,0,0,0))

sidenum <- 3:7

plot(Bali,usearrows=FALSE,vertex.cex=4,

displaylabels=F,vertex.sides=sidenum[rolecat])

par(op)

62 5 Effective Network Graphic Design

Fig. 5.6 Bali network with different node shapes

5.2.3 Node Size

Network node sizes are controlled by the vertex.cex optioninthestatnet

plot and gplot functions (similar to how sizes of graphic elements are controlled

in the base Rgraphics system). The overall sizes of the nodes should be set so that

the nodes are large enough to be distinguishable, but small enough that they do

not extensively overlap. In the following ‘Goldilocks’ example, we can see how

vertex.cex can be adjusted to ﬁnd an effective node size (Fig. 5.7).

op <- par(mar =c(0,0,2,0),mfrow=c(1,3))

plot(Bali,vertex.cex=0.5,main="Too small")

plot(Bali,vertex.cex=2,main="Just right")

plot(Bali,vertex.cex=6,main="Too large")

par(op)

Rather than setting the same overall size for every node, it is often useful to use

the node size in a network graphic to communicate some important quantitative

characteristic. For example, nodes vary in their positions in the overall network.

Some nodes are very central, while others are more peripheral. Chapter 7discusses

node prominence and centrality in more detail, but for now we will simply calculate

some node characteristics such that larger numbers indicate more central nodes.

To set this up, we will calculate three different measures of node centrality. Each

of these lines of code produces a vector of centrality measures for each node, and

larger numbers indicate greater centrality.

5.2 Design Elements 63

deg <- degree(Bali,gmode="graph")

deg

##[1]94915910399555559

## [16] 6 9

cls <- closeness(Bali,gmode="graph")

cls

## [1] 0.696 0.552 0.696 0.941 0.696 0.727 0.533

## [8] 0.696 0.696 0.571 0.571 0.571 0.571 0.571

## [15] 0.696 0.485 0.696

bet <- betweenness(Bali,gmode="graph")

bet

## [1] 2.333 0.333 1.667 61.167 1.667 6.167

## [7] 0.000 1.667 1.667 0.000 0.000 0.000

## [13] 0.000 0.000 1.667 0.000 1.667

Once you have this node-level vector of quantitative information, it can be used

to set the relative sizes of the nodes. This is done by using the same vertex.cex

Too small Just right Too large

Fig. 5.7 Adjusting overall node size

option as before, but instead of assigning a single number we assign the vector of

node information.

op <- par(mar =c(0,0,2,1),mfrow=c(1,2))

plot(Bali,usearrows=T,vertex.cex=deg,main="Raw")

plot(Bali,usearrows=FALSE,vertex.cex=log(deg),

main="Adjusted")

par(op)

However, as we can see by comparing the two panels in Fig. 5.8, the raw numbers

in the deg vector produce nodes that are much too large. They need to be adjusted,

64 5 Effective Network Graphic Design

and in this case we get usable sizes by taking the log of the deg values. This results

in node sizes where we can more easily see the nodes with higher degree relative to

nodes with lower degree.

Raw Adjusted

Fig. 5.8 Adjusting relative node size – Example 1

The next two examples show other types of adjustments that might be necessary

when setting relative node sizes. Using cls (closeness) we have the opposite prob-

lem from the previous example, where the nodes sizes start out too small. So an

appropriate adjustment is to multiply the original values (Fig. 5.9). The bet vector

(betweenness) provides a more complex example. First, the raw vector sizes vary

across several orders of magnitude (with one node with a size of 122.3). In addition,

some of the nodes have 0 for their bet values. These zeros would result in the nodes

being plotted with 0 size, so we need to handle this by adding 1 to the entire vector

before taking the square root (Fig. 5.10).

op <- par(mar =c(0,0,2,1),mfrow=c(1,2))

plot(Bali,usearrows=T,vertex.cex=cls,main="Raw")

plot(Bali,usearrows=FALSE,vertex.cex=4*cls,

main="Adjusted")

par(op)

op <- par(mar =c(0,0,2,1),mfrow=c(1,2))

plot(Bali,usearrows=T,vertex.cex=bet,main="Raw")

plot(Bali,usearrows=FALSE,vertex.cex=sqrt(bet+1),

main="Adjusted")

par(op)

The adjustments for relative node sizes can be tedious, although R does give you

complete control for how to adjust the sizes. The following function can be used to

save some time when ﬁguring out the best node sizes. The function rescale()

takes a vector of node characteristics (actually can be any numeric vector), and

rescales the values to ﬁt between the low and high values.

5.2 Design Elements 65

Raw Adjusted

Fig. 5.9 Adjusting relative node size – Example 2

Raw Adjusted

Fig. 5.10 Adjusting relative node size – Example 3

rescale <- function(nchar,low,high){

min_d <- min(nchar)

max_d <- max(nchar)

rscl <- ((high-low)*(nchar-min_d))/(max_d-min_d)+low

rscl

}

The next plot shows how the function works and rescales the raw degree values

for the Bali network to set the node sizes to vary from one to six (Fig. 5.11).

66 5 Effective Network Graphic Design

plot(Bali,vertex.cex=rescale(deg,1,6),

main="Adjusted node sizes with rescale function.")

Adjusted node sizes with rescale function.

Fig. 5.11 Rescaling the node size based on degree

5.2.4 Node Label

A network graphic is often more interesting and easier to interpret if nodes are

labelled so that the audience can see who or what makes up the network. This is

particularly helpful for smaller networks; if networks get too large then the labels

themselves may get in the way of the network information.

If a network object in statnet contains the special vertex attribute

vertex.names, then this can be used to automatically display node labels when

plotting. Other characteristics of the node labels can be controlled such as font size,

color, and distance from node (Fig. 5.12).

get.vertex.attribute(Bali,"vertex.names")

## [1] "Muklas" "Amrozi" "Imron" "Samudra"

## [5] "Dulmatin" "Idris" "Mubarok" "Husin"

5.2 Design Elements 67

## [9] "Ghoni" "Arnasan" "Rauf" "Octavia"

## [13] "Hidayat" "Junaedi" "Patek" "Feri"

## [17] "Sarijo"

op <- par(mar =c(0,0,0,0))

plot(Bali,displaylabels=TRUE,label.cex=0.8,

pad=0.4,label.col="darkblue")

par(op)

Muklas

Amrozi

Imron

Samudra

Dulmatin

Idris

Mubarok

Husin

Ghoni

Arnasan

Rauf

Octavia

Hidayat

Junaedi

Patek Feri

Sarijo

Fig. 5.12 Bali network with labelled nodes.

The automatic labels based on information stored in the vertex.names

attribute may not be the most important or useful information. For example, in the

case of the Bali network the actual names of the terrorists are not that interesting to

most viewers. Fortunately, you can use other text information to label the nodes. We

saw an example of this earlier in Fig. 5.3. In this case we are using the text stored

in the role vertex attribute to label the nodes. The key here is to use the label

option to specify what text vector to use for the labels (Fig. 5.13).

rolelab <- get.vertex.attribute(Bali,"role")

plot(Bali,usearrows=FALSE,label=rolelab,

displaylabels=T,label.col="darkblue")

68 5 Effective Network Graphic Design

Fig. 5.13 Bali network with role labels

5.2.5 Edge Width

If your network data include valued ties, or in general any quantitative information

that can be related to ties between nodes, then you can communicate that informa-

tion visually by altering the width of the displayed ties in a network graphic. For

example, the strength of friendship ties might be known, or the amount of money

that ﬂows between organizations in a directed network might be measured. In these

cases, thicker ties can denote greater strength or greater ﬂow (Fig. 5.14).

The Bali network includes a tie attribute called IC, which is a simple ﬁve-level

ordinal scale that was used to measure the amount of interaction between members

of the network. This attribute can be used to set the width of the ties in the network

visualization. In the example below the IC values are extracted from the stored edge

attribute, this allows us to transform the vector to better distinguish among the ﬁve

IC levels (by multiplying the vector by 1.5).

op <- par(mar =c(0,0,0,0))

IClevel <- Bali %e% "IC"

plot(Bali,vertex.cex=1.5,

edge.lwd=1.5*IClevel)

par(op)

5.2 Design Elements 69

5.2.6 Edge Color

While edge width can be set to communicate quantitative information about network

ties, the color of the edge can be set to communicate qualitative information about

the tie, similar to how node colors can set. For example, you could use different

colors of line graphics to distinguish between positive and negative ties in a social

network (Fig. 5.15).

The Bali network does not contain categorical or qualitative information stored in

an edge attribute, so here we create a random categorical vector to demonstrate how

to use different edge colors in a network graphic. For this example, we set up a color

palette that can be used to index the correct color choice, based on the categorical

edge vector. In this case blue will be used for edge type #1, red for edge type #2,

and green for edge type #3. This might reﬂect neutral ties (blue), negative ties (red),

and positive ties (green). (Also see Fig. 7.8 in Chap. 7for a more realistic example

of using different line colors.)

n_edge <- network.edgecount(Bali)

edge_cat <- sample(1:3,n_edge,replace=T)

linecol_pal <- c("blue","red","green")

Fig. 5.14 Bali network with edge widths indicating amount of interaction

70 5 Effective Network Graphic Design

plot(Bali,vertex.cex=1.5,vertex.col="grey25",

edge.col=linecol_pal[edge_cat],edge.lwd=2)

5.2.7 Edge Type

While edge width can be set to communicate quantitative information about network

ties, the type of the edge can be set to communicate qualitative information about

the ties. For example, you could use different types of line graphics to distinguish

between positive and negative ties in a social network.

The Bali network does not contain categorical or qualitative information stored

in an edge attribute, so here we create a random categorical vector to demonstrate

how to use different edge types in a network graphic. Here three different line types

are used (2 = dashed; 3 = dotted; 4 = dotdash). Also, the different line types do not

show up clearly using plot(),sogplot() is used here (Fig. 5.16).

Fig. 5.15 Bali network with different edge colors

n_edge <- network.edgecount(Bali)

edge_cat <- sample(1:3,n_edge,replace=T)

line_pal <- c(2,3,4)

5.2 Design Elements 71

gplot(Bali,vertex.cex=0.8,gmode="graph",

vertex.col="gray50",edge.lwd=1.5,

edge.lty=line_pal[edge_cat])

Although this works as intended, the resulting graphic is not very attractive and

(in my mind) is hard to interpret. Different line types should be used sparingly, and

probably only for very small networks with only two different line types. Most pub-

lished network graphics stick to color and maybe line width to distinguish among

different types of network ties.

Fig. 5.16 Bali network with different edge types

5.2.8 Legends

The examples above show how network graphic elements such as node color, node

shape, node size, edge type, edge width can be used to communicate important

characteristics of the network. As with other types of information graphics, it is

often useful to provide a legend so that the meaning of this information is clear to

the user.

The basic plotting functions contained in statnet do not have built-in func-

tionality for providing a network graphic legend. Fortunately, it is easy to use the

legend() function provided by basic Rto add a legend to a network graphic. In

the example below we replicate the network graphic from Fig. 5.5 but add a leg-

end to provide the node color key. We also scale the node sizes to reﬂect node

72 5 Effective Network Graphic Design

prominence based on degree. See ?legend for more details on how to use legends.

Figure 5.17, in fact, serves as a nice example of a carefully designed network graphic

that could be used as a ﬁnal product. It uses node size, node color, and a legend to

efﬁciently and clearly communicate the most important information contained in

the Bali network.

my_pal <- brewer.pal(5,"Dark2")

rolecat <- as.factor(get.vertex.attribute(Bali,"role"))

plot(Bali,vertex.cex=rescale(deg,1,5),

vertex.col=my_pal[rolecat])

legend("bottomleft",legend=c("BM","CT","OA","SB","TL"),

col=my_pal,pch=19,pt.cex=1.5,bty="n",

title="Terrorist Role")

Terrorist Role

Fig. 5.17 Bali network with legend

Chapter 6

Advanced Network Graphics

One eye sees, the other feels. (Paul Klee)

As the previous two chapters demonstrate, both statnet and igraph have

sophisticated plotting capabilities that can produce a very wide variety of net-

work graphics. However, these plotting functions cannot meet all of the analytic

or presentation needs. In particular, network scientists may wish to produce more

specialized network graphics. Also, while statnet and igraph excel at pro-

ducing high-quality publication ready network graphics, these graphics are static.

Fortunately, developers have started exploring how to take network graphics and

deliver them to web-based platforms where users can interact with the diagrams.

This chapter explores a few of these more specialized network graphic techniques,

as well as demonstrating how to produce some simple web-based interactive net-

work diagrams.

6.1 Interactive Network Graphics

One of the useful features of many other network analysis packages such as UCINet

and Pajek is the ability to produce network diagrams that are interactive at some

level. For example, in Pajek a network visualization can be produced in a sepa-

rate ‘Draw’ window, and then the user can interact with that window in various

ways to edit or change the network graphic. These capabilities can be very useful

for exploring the network, as well as ﬁne-tuning a network graphic for subsequent

dissemination.

Although R’s programmatic framework allows for detailed control over all the

elements of a network graphic, this is generally not made available to the user in an

interactive way. There are a few exceptions to this, as well as some new packages

that allow for creating interactive network diagrams that can be published to the

web. In this section a few of these options are demonstrated.

D.A. Luke, A User’s Guide to Network Analysis in R, Use R!,

DOI 10.1007/978-3-319-23883-8 6

74 6 Advanced Network Graphics

6.1.1 Simple Interactive Networks in

igraph

The igraph package includes the tkplot() function which supports simple

interactive network plots through a Tk graphics window. Only some features of

the network graphics can be modiﬁed. A typical use for this feature is to produce the

interactive graphic, adjust the node positions to improve the network layout, save the

node position coordinates and then use the coordinates to produce a ﬁnal (non-

interactive) network diagram. This work ﬂow is illustrated below with the Bali

network, see Chap. 8for a more in-depth example with these data.

library(intergraph)

library(igraph)

data(Bali)

iBali <- asIgraph(Bali)

Coord <- tkplot(iBali, vertex.size=3,

vertex.label=V(iBali)$role,

vertex.color="darkgreen")

# Edit plot in Tk graphics window before

# running next two commands.

MCoords <- tkplot.getcoords(Coord)

plot(iBali, layout=MCoords, vertex.size=5,

vertex.label=NA,vertex.color="lightblue")

6.1.2 Publishing Web-Based Interactive Network Diagrams

Instead of building interactive network graphics within Ritself, more people are

beginning to look at ways to produce interactive graphics that are published on the

Web, using frameworks like the D3 JavaScript library (http://http://d3js.

org/) and Shiny (http://shiny.rstudio.com/).

None of these approaches yet have come close to matching what a fully-developed

network graphics application such as Gephi can do. However, I anticipate that we

will be seeing rapid development of more R-connected approaches to web-based

network visualization in the next few years.

The networkD3 package is a small set of functions that can be used to build

simple interactive network graphics that can be displayed in shiny-aware documents

(i.e., RStudio) or in HTML web-pages. The following code shows how simple it is to

produce an interactive graphic. The ﬁrst set of lines will send a graphic to the Viewer

window if you run the commands within RStudio. The simpleNetwork()

function expects the network data in the form of an edgelist stored in a dataframe.

(The output from the examples in this section is not shown here, because it requires

RStudio or a web browser to view.)

6.1 Interactive Network Graphics 75

library(networkD3)

src <- c("A","A","B","B","C","E")

target <- c("B","C","C","D","B","C")

net_edge <- data.frame(src, target)

simpleNetwork(net_edge)

To save the interactive network to a freestanding HTML ﬁle, use the following

code.

net_D3 <- simpleNetwork(net_edge)

saveNetwork(net_D3,file ='Net_test1.html',

selfcontained=TRUE)

The output from simpleNetwork is so simple that it mainly is useful as a

proof-of-concept or tech demo. Slightly more sophisticated network graphics can be

produced using the forceNetwork() function. For this example, we are using

the Bali network again. The function expects data to be passed to it in two data

frames. The ‘links’ dataframe will have the network data in edgelist format. The

‘nodes’ dataframe will have the node id and properties of the nodes. Currently only

a categorical grouping variable is allowed. If the nodes have numeric ids, they must

start at 0. So, the main work to use the function is putting the data into the correct

format.

iBali_edge <- get.edgelist(iBali)

iBali_edge <- iBali_edge -1

iBali_edge <- data.frame(iBali_edge)

iBali_nodes <- data.frame(NodeID=as.numeric(V(iBali)-1),

Group=V(iBali)$role,

Nodesize=(degree(iBali)))

forceNetwork(Links = iBali_edge, Nodes = iBali_nodes,

Source ="X1",Target ="X2",

NodeID ="NodeID",Nodesize ="Nodesize",

radiusCalculation="Math.sqrt(d.nodesize)*3",

Group ="Group",opacity =0.8,

legend=TRUE)

Once again, this can be saved to an external ﬁle. Be careful, you will get an error

if you try to overwrite an existing ﬁle, even if it is not open in your browser.

net_D3 <- forceNetwork(Links = iBali_edge,

Nodes = iBali_nodes,

Source ="X1",Target ="X2",

NodeID ="NodeID",Nodesize ="Nodesize",

radiusCalculation="Math.sqrt(d.nodesize)*3",

Group ="Group",opacity =0.8,

legend=TRUE)

76 6 Advanced Network Graphics

saveNetwork(net_D3,file ='Net_test2.html',

selfcontained=TRUE)

The visNetwork package is a similar set of tools that uses the vis.js

javascript library (http://visjs.org/) to produce web-based interactive net-

work graphics.

This package also requires network data to be provided in a nodes data frame

and an edges data frame. The nodes data frame should include an id column, and

the edges data frame should have from and columns. Using the Bali network,

the following code sets up the data and produces a minimal example of an interactive

network graphic. Like in the previous example, this code produces an interactive

network in the Viewer window of RStudio.

library(visNetwork)

iBali_edge <- get.edgelist(iBali)

iBali_edge <- data.frame(from = iBali_edge[,1],

to = iBali_edge[,2])

iBali_nodes <- data.frame(id =as.numeric(V(iBali)))

visNetwork(iBali_nodes, iBali_edge, width ="100%")

The visNetwork package has a large number of options that can be used

to control the appearance of the network diagram, as well as for controlling how

the plot can be embedded in Shiny web applications. See the package help

ﬁle for more information, as well as a more in-depth demonstration of its capa-

bilities available at http://dataknowledge.github.io/visNetwork/.

The next code shows off some of these options.

iBali_nodes$group <- V(iBali)$role

iBali_nodes$value <- degree(iBali)

net <- visNetwork(iBali_nodes, iBali_edge,

width ="100%",legend=TRUE)

visOptions(net,highlightNearest =TRUE)

First, some of the display options are controlled by saving node or edge infor-

mation into the nodes or edges data frames. Here, the group variable stores the

‘role’ attribute, and the value variable is used to store the node sizes (in this case,

the degree). The visNetwork() and visOptions() functions are used to dis-

play the network, add a legend based on the grouping variable, set default colors

for each group, and then allow for the user to highlight individual nodes and their

immediate neighbors when clicking on a node in the diagram.

As before, these interactive plots will appear in a plot window if you are using

RStudio. Once the plot has been designed, it can be exported to a freestanding web-

page or embedded in other web platforms (e.g., with Shiny). This last example

shows how to save the plot in a separate web ﬁle, using the saveWidget()

function from the htmlwidgets package, which is installed when you install the

6.2 Specialized Network Diagrams 77

visNetwork package. This example adds a set of navigation buttons to the ﬁnal

network plot that allows moving the network and zooming in or out.

net <- visNetwork(iBali_nodes, iBali_edge,

width ="100%",legend=TRUE)

net <- visOptions(net,highlightNearest =TRUE)

net <- visInteraction(net,navigationButtons =TRUE)

library(htmlwidgets)

saveWidget(net, "Net_test3.html")

6.1.3 Statnet Web: Interactive

statnet

with

shiny

As evidence of the rapid development of interactive network tools, the Statnet devel-

opment team has recently published a web-based version of their Rnetwork analytic

tools using the shiny web application framework.

Statnet Web can be used by connecting directly to the shinyapps.io server

at https://statnet.shinyapps.io/statnetWeb. Or, the tools can be

run locally by installing the statnetWeb package. In addition to producing

basic network plots by selecting parameters and options from drop-down boxes,

statnetWeb can produce a variety of network statistics as well as ﬁt and test

ERGMs (see Chap. 11). Although web-based statnet does not give as much control

over or reproducibility of network analytic results as a programming approach does,

it is an impressive platform for quickly exploring network characteristics and will

be useful for teaching as well as disseminating network analytic results.

library(statnetWeb)

run_sw()

6.2 Specialized Network Diagrams

Traditionally, network diagrams are plotted to illustrate fundamental network and

node properties such as prominence (see Chap. 4). However, there are a number

of more specialized plotting techniques that can be used that are appropriate for

highlighting other important or interesting aspects of the networks. Three of these

approaches are demonstrated in this section: arc diagrams, chord diagrams, and

heatmaps.

78 6 Advanced Network Graphics

6.2.1 Arc Diagrams

Arc diagrams can be used when the positioning of nodes in a network is of less

interest than the pattern of ties. Here is a simple example of an arc diagram, using

the arcdiagram package. Note that this has to be installed using GitHub.

library(devtools)

install_github("gastonstat/arcdiagram")

The set-up for this example includes loading all the required libraries, then cre-

ating an edgelist object for the arcdiagram() function. For this example, we are

using the Simpsons dataset, which contains a set of (ﬁctitious) network data that

shows the primary interaction ties between 15 of the characters on the Simpsons

television show.

library(arcdiagram)

library(igraph)

library(intergraph)

data(Simpsons)

iSimp <- asIgraph(Simpsons)

simp_edge <- get.edgelist(iSimp)

A basic arc diagram can be produced with one function call (Fig. 6.1).

arcplot(simp_edge)

The arc diagram can be enhanced in a number of ways to highlight node and other

network characteristics. Here we deﬁne some subgroups in the network (1 = family,

2 = work, 3 = school, 4 = neighborhood) and use colors to distinguish the groups

(colors taken from a palette at colorbrewer2.org). Also, the degree of each

node is used to adjust its size (Fig. 6.2).

s_grp <- V(iSimp)$group

s_col =c("#a6611a","#dfc27d","#80cdc1","#018571")

cols =s_col[s_grp]

node_deg <- degree(iSimp)

arcplot(simp_edge, lwd.arcs=2,cex.nodes=node_deg/2,

labels=V(iSimp)$vertex.names,

col.labels="darkgreen",font=1,

pch.nodes=21,line=1,col.nodes = cols,

bg.nodes = cols, show.nodes =TRUE)

6.2 Specialized Network Diagrams 79

Fig. 6.1 Simpsons contact network

6.2.2 Chord Diagrams

Chord diagrams are a specialized type of information graphic that uses a circular

layout to display the interrelationships between data in a matrix. They have become

particularly popular in genetics research. Because network information can be org-

anized in matrices, chord diagrams are an interesting graphic option for network

plots. This is especially true for valued (weighted) and directed networks, where the

amount and direction of the ‘ﬂows’ are of interest.

The circlize package, by Zuguang Gu, implements a variety of circular

graphics, including chord diagrams. The package has a lot of features, giving

the user great control over the graphical appearance. The included vignette,

circular visualization of matrix is suggested reading.

In this example, we return to the network of the 2010 Netherlands World Cup

soccer team. Although Fig. 1.2 shows the basic pattern of passing ﬂows between the

eleven members of the team, it ignores the number of passes (stored in the vertex

attribute passes). Here we will create a chord diagram to further examine these

patterns.

The ﬁrst steps are to load the required packages and prepare the data. The main

requirement is to have the network data in the form of a sociomatrix, with the entries

corresponding to the strength or size of the tie if it is a valued network. The matrix

will also have to have names assigned for the rows and columns. (In this example

80 6 Advanced Network Graphics

Homer

Marge

Bart

Lisa

Maggie

Mr. Burns

Smithers

Pr. Skinner

Gr. Willie

Carl

Lenny

Moe

Milhouse

Ralph Wiggum

Ned Flanders

Fig. 6.2 Simpsons contact network – Version 2

we have an N×N matrix, so the names will be the same for rows and columns. The

circlize package can also be used for N×k matrices, so chord diagrams will

also be useful for 2-mode afﬁliation networks, such as those discussed in Chap. 9.)

library(statnet)

library(circlize)

data(FIFA_Nether)

FIFAm <- as.sociomatrix(FIFA_Nether,attrname='passes')

names <- c("GK1","DF3","DF4","DF5","MF6",

"FW7","FW9","MF10","FW11","DF2","MF8")

rownames(FIFAm) =names

colnames(FIFAm) =names

FIFAm

## GK1 DF3 DF4 DF5 MF6 FW7 FW9 MF10 FW11 DF2

## GK1 0 42 67 21 2 27 7 5 2 17

##DF3 30 044144215 8 7 1036

## DF4 38 43 0 57 18 11 7 21 1 7

6.2 Specialized Network Diagrams 81

## DF5 6 14 47 0 11 50 20 40 1 4

## MF6 9 28 25 10 0 41 28 37 14 34

## FW7 4 12 1 21 21 0 15 33 9 25

##FW9 0018712031167

## MF10 1 11 11 22 43 29 20 0 28 13

##FW113223761115 021

## DF2 29 38 8 3 45 38 10 18 26 0

## MF8 12 25 26 38 23 13 12 32 11 24

## MF8

## GK1 3

## DF3 29

## DF4 28

## DF5 42

## MF6 21

## FW7 18

## FW9 2

## MF10 21

## FW11 12

## DF2 15

## MF8 0

The sociomatrix reveals a number of ties that have very low numbers of passes.

To make the subsequent graphics a little easier to interpret we drop all ties with less

than ten passes.

FIFAm[FIFAm <10]<- 0

FIFAm

## GK1 DF3 DF4 DF5 MF6 FW7 FW9 MF10 FW11 DF2

## GK1 0 42 67 21 0 27 0 0 0 17

##DF3 30 044144215 0 0 1036

## DF4 38 43 0 57 18 11 0 21 0 0

## DF5 0 14 47 0 11 50 20 40 0 0

## MF6 0 28 25 10 0 41 28 37 14 34

## FW7 0 12 0 21 21 0 15 33 0 25

##FW9 0000012031160

## MF10 0 11 11 22 43 29 20 0 28 13

##FW110000001115 021

## DF2 29 38 0 0 45 38 10 18 26 0

## MF8 12 25 26 38 23 13 12 32 11 24

## MF8

## GK1 0

## DF3 29

## DF4 28

## DF5 42

## MF6 21

82 6 Advanced Network Graphics

## FW7 18

## FW9 0

## MF10 21

## FW11 12

## DF2 15

## MF8 0

With a sociomatrix that has names assigned, a basic chord diagram can be pro-

duced by a simple call to the chordDiagram() function (Fig. 6.3).

chordDiagram(FIFAm)

Chord diagrams can contain a lot of information, especially for larger networks,

so it is usually important to ﬁne tune the plot to highlight the most important infor-

mation. In this next plot, a number of options are used to make the graphic a little

easier to interpret. First, colors are set so that players in the same position (Forward,

Midﬁelder, etc.) have the same color. Then, because this is a directed network, ﬂows

(passes, in this case) go in both directions. The directional optionisusedso

that the departing passes start further away from outer circle, making it easier to see

the difference between passes sent and passes received. Finally, the order option

is used to sort the players by their position.

grid.col <- c("#AA3939",rep("#AA6C39",4),

rep("#2D882D",3),rep("#226666",3))

chordDiagram(FIFAm,directional =TRUE,

grid.col = grid.col,

order=c("GK1","DF2","DF3","DF4","DF5",

"MF6","MF8","MF10","FW7",

"FW9","FW11"))

In the resulting chord diagram (Fig. 6.4), it is much easier to see the patterns of

passes among the players. We can see that FW7 receives more than twice the number

of passes than the other two forwards. Similarly, we can see that the goalkeeper’s

favorite target is DF4, and that DF4 likes to pass frequently to DF5.

6.2.3 Heatmaps for Network Data

Heatmaps are another example of a specialized graphic that can be used for net-

works, especially valued or weighted networks. Here, a heatmap is produced to

highlight the players who are passing or receiving the most among the Netherlands

teammates.

First, a sociomatrix is created with the cells reﬂecting the tie weight, in this case

‘passes.’ Row and column names are deﬁned for the margin labels.

6.2 Specialized Network Diagrams 83

GK1

DF3

DF4

DF5

MF6

FW7

FW9

MF10

FW11

DF2

MF8

100

200

100

200

300

400

100

200

300

400

100

200

300

400

100

200

300

400

100

200

300

100

200

300

400

100

200

300

100

200

300

400

Fig. 6.3 Chord diagram of Netherlands 2010 soccer team, with all default options

data(FIFA_Nether)

FIFAm <- as.sociomatrix(FIFA_Nether,attrname='passes')

colnames(FIFAm) <- c("GK1","DF3","DF4","DF5",

"MF6","FW7","FW9","MF10",

"FW11","DF2","MF8")

rownames(FIFAm) <- c("GK1","DF3","DF4","DF5",

"MF6","FW7","FW9","MF10",

"FW11","DF2","MF8")

Once the data are set up, the heatmap is relatively easy to produce (Fig. 6.5). The

colorRampPalette() function is used to designate a color range that will be

used for the low and high ends of the values in the sociomatrix. (The color ranges

chosen here were taken from color chooser tools at paletton.com.) The network data

are directed, so it is important to remember that here the rows are the ‘passers’ and

the columns are the ‘receivers.’

84 6 Advanced Network Graphics

GK1

DF2

DF3

DF4

DF5

MF6

MF8

MF10

FW7

FW9

FW11

100

200

100

200

300

100

200

300

400

100

200

300

400

100

200

300

400

100

200

300

400

100

200

300

400

100

200

300

400

100

200

300

100

Fig. 6.4 Chord diagram of Netherlands 2010 soccer team, with advanced options

palf <- colorRampPalette(c("#669999","#003333"))

heatmap(FIFAm[,11:1],Rowv =NA,Colv =NA,col =palf(60),

scale="none",margins=c(11,11))

The heatmap also shows the same pattern of heavy passers as Fig. 6.4. The dark-

est square is for the passes from the goalkeeper to DF4.

6.3 Creating Network Diagrams with Other R Packages

6.3.1 Network Diagrams with

ggplot2

Although ggplot2 is not designed to handle all of the requirements of a full-

ﬂedged network visualization package, some of its advanced graphics capabilities

can be used to create specialized network plotting routines. The following example

6.3 Creating Network Diagrams with Other R Packages 85

MF8

DF2

FW11

MF10

FW9

FW7

MF6

DF5

DF4

DF3

GK1

DF3

DF4

DF5

MF6

FW7

FW9

MF10

FW11

DF2

MF8

Fig. 6.5 Heatmap of Netherlands 2010 soccer team number of passes

is based on code developed by David Sparks, and posted on the blog he runs with

his colleague Christopher DeSante, is.R() (http://is-r.tumblr.com).

The edgeMaker() function and supporting code can be used to create att-

ractive and functional plots of directed networks using ‘tapered-intensity-curved’

edges. The bulk of the work is done by the edgeMaker() function which creates

the curved ties between each connected dyad.

edgeMaker <- function(whichRow,len=100,curved =TRUE){

fromC <- layoutCoordinates[adjacencyList[whichRow,1],]

toC <- layoutCoordinates[adjacencyList[whichRow,2],]

graphCenter <- colMeans(layoutCoordinates)

bezierMid <- c(fromC[1], toC[2])

distance1 <- sum((graphCenter -bezierMid)ˆ2)

if(distance1 <sum((graphCenter -c(toC[1],

fromC[2]))ˆ2)){

bezierMid <- c(toC[1], fromC[2])

}

bezierMid <- (fromC +toC +bezierMid) /3

if(curved == FALSE){bezierMid <- (fromC +toC) /2}

edge <- data.frame(bezier(c(fromC[1], bezierMid[1],

toC[1]),

c(fromC[2], bezierMid[2],

toC[2]),

evaluation = len))

86 6 Advanced Network Graphics

edge$Sequence <- 1:len

edge$Group <- paste(adjacencyList[whichRow, 1:2],

collapse =">")

return(edge)

}

In addition to the core sna and ggplot2 packages, the Hmisc package is used

which provides the bezier() functionusedbyedgeMaker.

library(sna)

library(ggplot2)

library(Hmisc)

As has been typical with the examples in this chapter, the network data has to

be transformed to an edgelist format prior to using the plotting functions. For this

example, we also drop the ties that have the associated weight ‘passes’ less than 10.

Finally, the edgeMaker function expects the edgelist object to be named ‘adjacen-

cyList.’

data(FIFA_Nether)

fifa <- FIFA_Nether

fifa.edge <- as.edgelist.sna(fifa,attrname='passes')

fifa.edge <- data.frame(fifa.edge)

names(fifa.edge)[3]<- "value

fifa.edge <- fifa.edge[fifa.edge$value >9,]

adjacencyList <- fifa.edge

Now, we use edgeMaker to create the curved edges. Also, gplot (from sna)

is called once to store the layout coordinates for the ggplot2 function. (This

means that any set of coordinates can be fed to ggplot2.)

layoutCoordinates <- gplot(network(fifa.edge))

allEdges <- lapply(1:nrow(fifa.edge),

edgeMaker, len =500,curved =TRUE)

allEdges <- do.call(rbind, allEdges)

Before producing the plot, we create an empty ggplot2 theme. This is used to

clean up after producing the plot.

new_theme_empty <- theme_bw()

new_theme_empty$line <- element_blank()

new_theme_empty$rect <- element_blank()

new_theme_empty$strip.text <- element_blank()

new_theme_empty$axis.text <- element_blank()

new_theme_empty$plot.title <- element_blank()

new_theme_empty$axis.title <- element_blank()

new_theme_empty$plot.margin <- structure(c(0,0,-1,-1),

6.3 Creating Network Diagrams with Other R Packages 87

unit ="lines",

valid.unit =3L,

class ="unit")

And now the ﬁnal step is to create the plot using ggplot(). Familiarity with

ggplot2 will help in understanding this code. The scale colour gradient

option controls the intensity of the gradient, and the scale size option controls

the amount of the taper (Fig. 6.6).

zp1 <- ggplot(allEdges)

zp1 <- zp1 +geom_path(aes(x=x,y=y,group = Group,

colour=Sequence, size=-Sequence))

zp1 <- zp1 +geom_point(data =

data.frame(layoutCoordinates),

aes(x=x,y= y),

size =4,pch =21,

colour ="black",fill ="gray")

zp1 <- zp1 +scale_colour_gradient(low =gray(0),

high =gray(9/10),

guide ="none")

zp1 <- zp1 +scale_size(range =c(1/10,1.5),

guide ="none")

zp1 <- zp1 +new_theme_empty

print(zp1)

Fig. 6.6 Netherlands 2010 passing network with curved ties

Part III

Description and Analysis

Chapter 7

Actor Prominence

...we’re now tied up with human beings, tied to you and forced

to go on with this adventure according to the laws of visibility.

(Jean Genet)

7.1 Introduction

Networks are interesting because of their speciﬁc structural patterns, and how those

structures affect the members of the network. Stated more simply, networks affect

their members based on where those members are located in the networks. A person

who is connected to many other members of a network is likely to view the rest

of the network quite differently from somebody who is relatively isolated from the

other members.

Network analysis provides many tools for viewing, analyzing, and assessing

the locations of individual nodes and ties. This is often the ﬁrst type of network

analysis that is performed once network data are obtained, beyond simple network

description.

By examining the location of individual network members, we can assess the

prominence of those members. An actor is prominent if the ties of the actor make

that actor visible to the other members in the network (Knoke and Burt 1983). In

the rest of this chapter, we will cover a number of the most common ways to assess

network member prominence. For non-directed networks we will look at centrality;

where we view a central actor as one who is involved in many (direct or indirect) ties.

For directed networks, prominence is usually referred to as prestige; a prestigious

actor is one who is the object of extensive ties. This chapter will also cover how

individual node-level measures of centrality and prestige can be aggregated into

network-level centralization measures. An example of how to report the results of

prominence analysis will be presented. Finally, there will be a short discussion of

identifying cutpoints and bridges in networks. These are technically not measures

of prominence, but are simple locational properties of individual nodes or ties, and

as such are somewhat similar to the rest of the chapter’s subject matter.

D.A. Luke, A User’s Guide to Network Analysis in R, Use R!,

DOI 10.1007/978-3-319-23883-8 7

92 7 Actor Prominence

7.2 Centrality: Prominence for Undirected Networks

It makes intuitive sense that a network member who is connected to many other

members of the network is in a prominent position. For non-directed networks, we

will say that this type of actor has high centrality, or that it is in a central position.

However, there are a number of ways of operationalizing this type of prominence.

In fact, there are dozens of centrality statistics available to the network analyst.

To see how we can come up with different types of centrality measures, consider

the example network displayed in Fig. 7.1, based on the following simple socioma-

trix, called net mat.

## abcdefghij

##a0110000000

##b1010000000

##c1101101000

##d0010100000

##e0011010000

##f0000101000

##g0010010100

##h0000001011

##i0000000100

##j0000000100

Which node is most central? Nodes cand gare both positioned in the center of

the graph, but as we learned in Chap. 4, the location of nodes in network graphics

may or may not hold any particular meaning. However, node cis directly connected

to more network members than any other node, so in that sense we could view cas

a central node. Alternatively, node gdoes not have as many direct network ties, but

it is positioned in such a way that it connects two different parts of the network. In

particular, the only way that information from nodes h, i, and jgets to the rest of the

network is through node g. Finally, even though node gis only directly connected

to two other nodes, it is positioned so that it is fairly close to every other node in the

network. Speciﬁcally, node gcan reach every other node in only one or two steps.

That is, node gis connected to the rest of the network by paths of length one or two.

So, in these two very different senses, node gcan also be thought of as a central

node.

In the next three sections, we will cover the three most commonly used measures

of centrality.

7.2 Centrality: Prominence for Undirected Networks 93

Fig. 7.1 Network graph example to demonstrate concepts of prominence

7.2.1 Three Common Measures of Centrality

7.2.1.1 Degree Centrality

The simplest measure of centrality by far is based on the notion that a node that

has more direct ties is more prominent than nodes with fewer or no ties. Degree

centrality thus, is simply the degree of each node. We ﬁrst introduced node degree

in Chap. 2. The degree of a node is the number of ties it has with other nodes.

Following the notation of Wasserman and Faust (1994), degree centrality is

deﬁned as:

CD(ni)=d(ni)

The network in Fig. 7.1 is simple enough that we could count up the node degrees

by hand. However, here is how degree centrality can be calculated in statnet,

assuming that we have the data stored in a network object called net.

net <- network(net_mat)

net %v% 'vertex.names'

## [1] "a" "b" "c" "d" "e" "f" "g" "h" "i" "j"

degree(net, gmode="graph")

## [1]2252323311

The ﬁrst line of code simply reminds you of the names of the nodes and their

order. The degree() function calculates and returns the degree centrality scores

for each node. The gmode option tells the function to treat the network object as a

non-directed network (graph). (This option needs to be used, even if the network is

created and stored as a non-directed network.)

The results conﬁrm what we had already suggested above. Node chas the highest

degree centrality. It is connected to ﬁve other nodes in the network, more than any

other node.

94 7 Actor Prominence

7.2.1.2 Closeness Centrality

Instead of examining only the direct connections of the nodes, we can focus on how

close each node is to every other node in a network. This leads to the concept of

closeness centrality, where nodes are more prominent to the extent they are close to

all other nodes in the network. Here is the relatively more complicated equation for

closeness centrality,

CC(ni)=g

∑

j=1

d(ni,nj)−1

where dis the path distance between two nodes. Closeness centrality, then, is the

inverse of the sum of all the distances between node iand all the other nodes in the

network.

closeness(net, gmode="graph")

## [1] 0.409 0.409 0.600 0.429 0.450 0.450 0.600

## [8] 0.474 0.333 0.333

This tells us that nodes cand gare tied with the highest closeness.

7.2.1.3 Betweenness Centrality

Betweenness centrality measures the extent that a node sits ‘between’ pairs of other

nodes in the network, such that a path between the other nodes has to go through

that node. A node with high betweenness is prominent, then, because that node is in

a position to observe or control the ﬂow of information in the network. The equation

for betweenness centrality is

CB(ni)=∑

j<k

gjk(ni)/gjk

where gjk is the geodesic between nodes jand k. (A geodesic is the shortest path

between two nodes.) gjk(ni)is the number of geodesics between nodes jand kthat

contain node i.

betweenness(net, gmode="graph")

## [1] 0.0 0.0 20.0 0.0 2.5 2.0 19.5 15.0 0.0

## [10] 0.0

This shows that node chas the highest betweenness score, with nodes gand h

not far behind. These quick examples show that different measures of centrality will

emphasize different aspects of the prominence of nodes in a network.

7.2 Centrality: Prominence for Undirected Networks 95

7.2.2 Centrality Measures in R

R can handle many different measures of centrality and prestige. See the accom-

panying table for a list of the measures currently included in the statnet and

igraph packages (Table 7.1).

As we can see, R provides a wide variety of ways to examine the centrality and

prestige of individual actors in a network. The choice of which measure of centrality

or prestige to use is driven in part by the type of network data you have; in particular,

whether the network is directed or not. However as suggested in Sect. 7.2.1,the

choice should be primarily driven by what type of information is provided by the

particular prominence measure.

That being said, it is also useful to keep in mind that in many real-world social

networks there is a great deal of overlap in the various centrality and prestige mea-

sures. Nodes that are identiﬁed as highly central using eigenvector centrality are also

likely to be identiﬁed as central with other measures, especially those most closely

related to eigenvector centrality (e.g., Bonacich power). We can illustrate this by

showing the correlations among a set of centrality measures available in statnet

applied to the DHHS Collaboration network.

Measures statnet igraph

Degree degree() degree()

Closeness closeness() closeness()

Betweenness betweenness() betweenness()

Eigenvector evcent() evcent()

Bonacich power bonpow() bonpow()

Flow betweenness ﬂowbet()

Load loadcent()

Information infocent()

Stress stresscent()

Harary graph graphcent()

Bonacich alpha alpha.centrality()

Kleinberg authority auth.score()

Kleinberg hub hub.score()

PageRank page.rank()

Table 7.1 Prominence measures available in statnet and igraph

data(DHHS)

df.prom <- data.frame(

deg =degree(DHHS),

cls =closeness(DHHS),

btw =betweenness(DHHS),

evc =evcent(DHHS),

inf =infocent(DHHS),

flb =flowbet(DHHS)

)

96 7 Actor Prominence

cor(df.prom)

## deg cls btw evc inf flb

## deg 1.000 0.973 0.750 0.972 0.902 0.944

## cls 0.973 1.000 0.787 0.934 0.890 0.941

## btw 0.750 0.787 1.000 0.600 0.485 0.884

## evc 0.972 0.934 0.600 1.000 0.940 0.843

## inf 0.902 0.890 0.485 0.940 1.000 0.773

## flb 0.944 0.941 0.884 0.843 0.773 1.000

7.2.3 Centralization: Network Level Indices of Centrality

Centrality and prestige are characteristics of nodes in a network, based on the posi-

tion of the node in the overall network. The variability of the individual centrality

scores in a network can be very informative. For example, consider the following

two extreme examples: a star graph, and a circle graph (Fig. 7.2).

dum1 <- rbind(c(1,2),c(1,3),c(1,4),c(1,5))

star_net <- network(dum1,directed=FALSE)

dum2 <- rbind(c(1,2),c(2,3),c(3,4),c(4,5),c(5,1))

circle_net <- network(dum2,directed=FALSE)

par(mar=c(4,4,.1,.1))

my_pal <- brewer.pal(5,"Set2")

gplot(star_net,usearrows=FALSE,displaylabels=FALSE,

vertex.cex=2,

vertex.col=my_pal[1],

edge.lwd=0,edge.col="grey50",xlab="Star Graph")

gplot(circle_net,usearrows=FALSE,displaylabels=FALSE,

vertex.cex=2,

vertex.col=my_pal[3],

edge.lwd=0,edge.col="grey50",xlab="Circle Graph")

In statnet, centralization is calculated using the centralization()

function. The function accepts a name of an existing centrality or prestige function,

and returns the appropriate network-level centralization score. Note that

despite its name and the information presented in the help ﬁle for the function,

centralization() can be used for directed graphs.

Using the star and circle graphs, we can see that every node has the same cen-

trality score for the circle graph, leading to a minimum centralization score. The

star graph shows the opposite pattern, where there is high variability between the

node-level centrality scores, leading to higher centralization scores.

7.2 Centrality: Prominence for Undirected Networks 97

Star Graph Circle Graph

Fig. 7.2 Centralization extreme examples

closeness(circle_net)

## [1] 0.667 0.667 0.667 0.667 0.667

centralization(circle_net,closeness)

## [1] 0

closeness(star_net)

## [1] 1.000 0.571 0.571 0.571 0.571

centralization(star_net,closeness)

## [1] 0.536

7.2.4 Reporting Centrality

All centrality and prestige functions in statnet (as well as igraph) produce a

vector of node-level scores, one for each actor in the network. Using the Bali terror-

ist network, we can see that centrality varies widely across the network members.

data(Bali)

str(degree(Bali))

## num [1:17] 18 8 18 30 18 20 6 18 18 10 ...

summary(degree(Bali))

## Min. 1st Qu. Median Mean 3rd Qu. Max.

## 6.0 10.0 18.0 14.8 18.0 30.0

These scores can be examined individually, but for both analysis and reporting, it

is usually more informative to examine patterns of prominence across nodes, across

different prominence measures, and even across different networks. In this section

98 7 Actor Prominence

we take a more in-depth look at the centrality of the actors in the Bali terrorist

network (Fig. 7.3).

data(Bali)

my_pal <- brewer.pal(5,"Set2")

rolecat <- Bali %v% "role"

gplot(Bali,usearrows=FALSE,displaylabels=TRUE,

vertex.col=my_pal[as.factor(rolecat)],

edge.lwd=0,edge.col="grey25")

legend("topright",legend=c("BM","CT","OA","SB",

"TL"),col=my_pal,pch=19,pt.cex=2)

If the network is small enough, it can be useful to examine the individual node-

level prominence scores. Table 7.2 shows the individual scores for each of three

common centrality measures: degree, closeness, and betweenness.

Muklas

Amrozi

Imron

Samudra

Dulmatin

Idris

Mubarok

Husin

Ghoni

Arnasan

Rauf

Octavia

Hidayat

Junaedi

Patek

Feri

Sarijo

Fig. 7.3 Interactions among the 17 members of the 2002 Bali terrorist network

data(Bali)

df.prom2 <- data.frame(

degree =degree(Bali),

closeness =closeness(Bali),

betweenness =betweenness(Bali)

)

row.names(df.prom2) <- Bali %v% "vertex.names"

df.promsort <- df.prom2[order(-df.prom2$degree),]

cd <- centralization(Bali,degree)

cc <- centralization(Bali,closeness)

cb <- centralization(Bali,betweenness)

7.2 Centrality: Prominence for Undirected Networks 99

df.promsort <- rbind(df.promsort,c(cd,cc,cb))

row.names(df.promsort)[18]<- "\\emph{Centralization}"

Degree Closeness Betweenness

Samudra 30.00 0.94 122.33

Idris 20.00 0.73 12.33

Muklas 18.00 0.70 4.67

Imron 18.00 0.70 3.33

Dulmatin 18.00 0.70 3.33

Husin 18.00 0.70 3.33

Ghoni 18.00 0.70 3.33

Patek 18.00 0.70 3.33

Sarijo 18.00 0.70 3.33

Feri 12.00 0.48 0.00

Arnasan 10.00 0.57 0.00

Rauf 10.00 0.57 0.00

Octavia 10.00 0.57 0.00

Hidayat 10.00 0.57 0.00

Junaedi 10.00 0.57 0.00

Amrozi 8.00 0.55 0.67

Mubarok 6.00 0.53 0.00

Centralization 0.54 0.33 0.50

Table 7.2 Centrality of the 17 members of the 2002 Bali terrorist network

As described in Chap. 5, nodes can be sized according to any informative quanti-

tative characteristic of the actors. This can be non-network information such as age

or weight. More useful here is to use information from the network itself; in this

case the centrality scores for each node.

This can easily be done using the network plotting options. In fact, only one addi-

tional parameter (vertex.cex) needs to be passed to gplot(). This parameter

can be a constant, in which case it simply controls the overall size of each vertex

in the graph. However, you can also pass it a vector of numeric scores. All of the

node-level prominence measures return a numeric vector, so that is what can be used

to scale node size based on centrality or prestige.

The only tricky issue is that R reads the raw numbers passed to vertex.cex,

and these numbers are often too small or too large. Typically, you will need to

play around with some type of scaling factor to ensure that the graphic is inter-

pretable. (See Chap. 5for a more detailed treatment of node sizing and scaling.) The

following two ﬁgures illustrate this. First, the normalized degree centrality for the

Bali network is calculated (option rescale=TRUE). This rescales the raw degree

scores so that they all fall between 0 and 1. The ﬁrst ﬁgure shows that the normal-

ized degree scores are too small. The second graph uses the same information, but

the vertex.cex parameter is multiplied by 20 so that the relative differences between

the node sizes can be seen (Fig. 7.4).

100 7 Actor Prominence

deg <- degree(Bali,rescale=TRUE)

op <- par(mfrow=c(1,2))

gplot(Bali,usearrows=FALSE,displaylabels=FALSE,

vertex.cex=deg,

vertex.col=my_pal[as.factor(rolecat)],

edge.lwd=0,edge.col="grey25",

main="Too small")

gplot(Bali,usearrows=FALSE,displaylabels=FALSE,

vertex.cex=deg*20,

vertex.col=my_pal[as.factor(rolecat)],

edge.lwd=0,edge.col="grey25",

main="A little better")

par(op)

Too small A little better

Fig. 7.4 Comparison of two approaches to sizing vertices by degree centrality

A network graphic that includes node-level prominence information can be an

effective analysis and communication tool. The overall structure of the network can

be made clear, as well as the importance of individual positions. Figure 7.5 is a ﬁnal

version of the Bali network graphic that combines node-level categorical informa-

tion (denoted by vertex color) with node-level quantitative information (denoted by

vertex size).

deg <- degree(Bali,rescale=TRUE)

gplot(Bali,usearrows=FALSE,displaylabels=TRUE,

vertex.cex=deg*12,

vertex.col=my_pal[as.factor(rolecat)],

edge.lwd=0.5,edge.col="grey75")

legend("topright",legend=c("BM","CT","OA","SB","TL"),

col=my_pal,pch=19,pt.cex=2)

7.3 Cutpoints and Bridges 101

Muklas

Amrozi

Imron

Samudra

Dulmatin

Idris

Mubarok

Husin

Ghoni

Arnasan

Rauf

Octavia

Hidayat

Junaedi

Patek

Feri

Sarijo

Fig. 7.5 Bali network with nodes sized according to degree centrality

7.3 Cutpoints and Bridges

There are two additional concepts from graph theory that can be useful tools when

assessing locational properties of individual nodes or ties. The ﬁrst is a cutpoint,

which is deﬁned as a node that, if dropped, would increase the number of com-

ponents in the network. In many types of networks cutpoints thus occupy important

positions connecting different parts of the network. If they were dropped, that would

result in two subsets of actors that would not be able to communicate with each other

(Fig. 7.6).

You can use the cutpoint() function in statnet to quickly identify any

cutpoints in a network. (For a directed network, you need to specify whether

you are using a ‘weak’ or ‘strong’ component rule for identifying cutpoints. See

?cutpoints for further information.)

cpnet <- cutpoints(net,mode="graph",

return.indicator=TRUE)

gplot(net,gmode="graph",vertex.col=cpnet+2,coord=coords,

jitter=FALSE,displaylabels=TRUE)

So, in addition to the two central nodes (cand g) we had identiﬁed earlier, we can

see that his also a cutpoint. Although simple to see in this example, we can conﬁrm

the nodes as cutpoints in a few different ways (Fig. 7.7).

net2 <- net

components(net2)

## [1] 1

102 7 Actor Prominence

delete.vertices(net2,7)

components(net2)

## [1] 2

gplot(net2,gmode="graph",vertex.col=2,

coord=coords[-7,],jitter=FALSE,displaylabels=TRUE)

Bridges are the edge equivalent to cutpoints. That is, an edge is a bridge if remov-

ing it will split one component into two. There is no bridge identiﬁcation function

built into statnet, but it is relatively easy to create a function that will detect

bridges. This function takes a statnet directed or non-directed network, and ex-

amines each tie to see if removing it changes the component count. A logical vector

with length equal to the number of ties is returned indicating which ties are bridges.

Fig. 7.6 Example graph with identiﬁed cutpoints

bridges <- function(dat,mode="graph",

connected=c("strong","weak")) {

e_cnt <- network.edgecount(dat)

if (mode == "graph"){

cmp_cnt <- components(dat)

b_vec <- rep(FALSE,e_cnt)

for(i in 1:e_cnt){

dat2 <- dat

delete.edges(dat2,i)

b_vec[i] <- (components(dat2) != cmp_cnt)

}

else {

cmp_cnt <- components(dat,connected=connected)

b_vec <- rep(FALSE,e_cnt)

for(i in 1:e_cnt){

dat2 <- dat

7.3 Cutpoints and Bridges 103

Fig. 7.7 Example graph with one cutpoint dropped

delete.edges(dat2,i)

b_vec[i] <- (components(dat2,connected=connected)

!= cmp_cnt)

}

return(b_vec)

}

Once the function has been deﬁned, we can use it directly:

bridges(net)

## [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE

## [8] FALSE FALSE FALSE FALSE FALSE FALSE FALSE

## [15] FALSE FALSE FALSE FALSE TRUE TRUE TRUE

## [22] TRUE TRUE TRUE

This shows us that there are three ties that are bridges in the example network.

We can also use the bridges function similarly to the cutpoints function in a graphic

to display which edges are bridges (Fig. 7.8).

brnet <- bridges(net)

gplot(net,gmode="graph",vertex.col="red",

edge.col=brnet+2,coord=coords,

jitter=FALSE,displaylabels=TRUE)

104 7 Actor Prominence

Fig. 7.8 Example graph with identiﬁed bridges

Chapter 8

Subgroups

Our young people are faced by a series of different groups

which believe different things and advocate different practices,

and to each of which some trusted friend or relative may belong.

(Margaret Mead)

8.1 Introduction

The social systems contained in networks often exhibit complex structures. For

example, in his classic The strength of weak ties, Granovetter (1973) suggested that

many social networks are made up of relatively densely connected subgroups (e.g.,

friendship subnetworks) that are themselves only connected via less common ties

(e.g., between acquaintances). It then follows that it will be important to be able to

deﬁne and identify such subgroups. Many disciplines have theories that assume that

larger social systems are made up of distinguishable subgroups, for example soci-

ologists consider social classes; psychologists examine small group behavior, and

public health examine health disparities between different social groups.

This chapter covers a number of techniques available within R to identify and

examine subgroups that may be contained in larger social networks. The igraph

package is used extensively in this chapter, because of the depth of its coverage of

subgroup and community detection techniques.

At times, it may not be necessary to use speciﬁc subgroup techniques. For

example, reconsider Moreno’s sociogram that we examined in Chap. 2(Fig. 8.1).

Here, it is self-evident that the network is made up of two primary groups, even if

we did not know beforehand that this depicts a primary school class.

However, in most real-world social networks the subgroup structure is not as

clear, if it even exists at all. Figure 8.2 shows a more realistic network, where the

there is a hint of some subgroup structure, but more systematic analysis will be

required to reveal it. The color coding and labels suggest that the there may be some

cohesion among members from the same DHHS agency, but it is not crystal clear.

D.A. Luke, A User’s Guide to Network Analysis in R, Use R!,

DOI 10.1007/978-3-319-23883-8 8

105

106 8 Subgroups

Fig. 8.1 Moreno sociogram, showing two subgroups

8.2 Social Cohesion

One way to think about network subgroups is through social cohesion. Cohesive

subgroups are sets of actors that are tied together through frequent, strong, and direct

ties (Wasserman and Faust 1994). This approach is so intuitive that it led to a number

of the earliest techniques for identifying network subgroups.

8.2 Social Cohesion 107

8.2.1 Cliques

Cliques are one of the simplest types of cohesive subgroups, and because of their

straightforward deﬁnition are also one of the easiest types to understand. A clique is

a maximally complete subgraph; that is, it is a subset of nodes that have all possible

ties among them.

ACF-1

ACF-2

AHRQ-1

AHRQ-2

AHRQ-3

AHRQ-4

CDC-1

CDC-2

CDC-3

CDC-4

CDC-5

CDC-6

CDC-7

CDC-8

CDC-9

CDC-10

CDC-11

CDC-12

CMS-1

CMS-2

FDA-1

FDA-2

HRSA-1

HRSA-2

HRSA-3

IHS-1

IHS-2

NIH-1

NIH-2

NIH-3

NIH-4

NIH-5

NIH-6

NIH-7

NIH-8

NIH-9

NIH-10

NIH-11

NIH-12

NIH-13

NIH-14

NIH-15

NIH-16

OGC-1

OGC-2 OGC-3

OS-1

OS-2

OS-3

OS-4

OS-5

SAMHSA-1

SAMHSA-2

SAMHSA-3

Fig. 8.2 Collaboration ties among DHHS agencies

Consider the example graph in Fig. 8.3. There are two cliques in this graph:

A,B,C,D and E,F,G. (Technically, connected dyads also are cliques, but typically

only cliques of size 3 or larger are of interest. Also, by deﬁnition any clique of size

kwill also contain all the cliques sized k-1,k-2, etc.)

library(igraph)

clqexmp <- graph.formula(A:B:C:D--A:B:C:D,D-E,E-F-G-E)

The following commands demonstrate how to get information about any cliques

in a network. Despite what the name suggests, clique.number() does not

return the number of cliques, but the size of the largest clique. To get a list of all

the cliques, constrained by a minimum or maximum size, use cliques(). When

there are a large number of cliques in a network, maximal.cliques() may be

more useful. Finally, as the name suggests, largest.cliques() will ﬁnd all of

the largest cliques in a network.

108 8 Subgroups

Fig. 8.3 Example graph with two cliques

clique.number(clqexmp)

## [1] 4

cliques(clqexmp, min=3)

## [[1]]

## + 3/7 vertices, named:

## [1] A B C

## [[2]]

## + 3/7 vertices, named:

## [1] A B D

## [[3]]

## + 3/7 vertices, named:

## [1] A C D

## [[4]]

## + 3/7 vertices, named:

## [1] B C D

## [[5]]

## + 3/7 vertices, named:

## [1] E F G

## [[6]]

## + 4/7 vertices, named:

## [1] A B C D

maximal.cliques(clqexmp,min=3)

8.2 Social Cohesion 109

## [[1]]

## + 3/7 vertices, named:

## [1] E F G

## [[2]]

## + 4/7 vertices, named:

## [1] A B D C

largest.cliques(clqexmp)

## [[1]]

## + 4/7 vertices, named:

## [1] D A B C

Note that the latter three functions return lists of vertex ids. When the igraph

object has vertex names, the following syntax shows how names rather than ids can

be displayed.

V(clqexmp)[unlist(largest.cliques(clqexmp))]

## + 4/7 vertices, named:

## [1] D A B C

Cliques, however, have two major disadvantages that reduce their utility in real-

world social network analysis. First, a clique is a very conservative deﬁnition of a

cohesive subgroup. Consider a subgraph made up of seven vertices. To be a clique,

all of the 21 possible ties must exist between all seven members. If only one is miss-

ing, then the seven vertices will not belong to one clique, even though the density of

these seven vertices (20/21 = 0.95) would suggest that this is a cohesive subgroup.

A consequence of this fragility is the second major issue of cliques: they simply

are not very common in larger social networks. Table 8.1 presents some simple

simulation results demonstrating the rarity of cliques. Four random networks were

created with 25, 50, 100, and 500 nodes. For each network, the average degree was

constrained to approximately 6. The table shows that the number of cliques remains

roughly constant, even as the network size increases dramatically. Furthermore, the

cliques remain small in size. (See Chap. 10 for more information on random graph

models such as erdos.renyi.game.)

g25 <- erdos.renyi.game(25,75,type="gnm")

g50 <- erdos.renyi.game(50,150,type="gnm")

g100 <- erdos.renyi.game(100,300,type="gnm")

g500 <- erdos.renyi.game(500,1500,type="gnm")

nodes <- c(25,50,100,500)

lrgclq <- c(clique.number(g25),clique.number(g50),

clique.number(g100),clique.number(g500))

numclq <- c(length(cliques(g25,min=3)),

length(cliques(g50,min=3)),

110 8 Subgroups

length(cliques(g100,min=3)),

length(cliques(g500,min=3)))

clqinfo <- data.frame(Nodes=nodes,Largest=lrgclq,

Number=numclq)

Nodes Largest Number

25 3 27

50 4 47

100 3 35

500 3 38

Table 8.1 Demonstration of clique characteristics

8.2.2 k-Cores

Partly because of the rarity of cliques in observed social networks, a number of

variations on the clique idea have been proposed. A popular variation is the k-core.

Ak-core is a maximal subgraph where each vertex is connected to at least kother

vertices in the subgraph. k-cores have a number of advantages: they are nested (every

member of a 4-core is also a member of a 3-core, and so on), they do not overlap,

and they are easy to identify.

An analysis of k-cores typically proceeds by ﬁrst identifying the entire k-core set,

then doing visual examination of the k-core structures, possibly followed by closer

examination of an individual k-core level (e.g., the 6-core).

For an example of how to examine k-cores in a social network we will use

the DHHS dataset. This dataset shows collaboration ties among tobacco control

experts working across various institutes and agencies within the Department of

Health and Human Services in 2005 (Leischow, Luke, et al. 2010). This dataset is a

statnet network object, so we start by translating it to an igraph object using

the intergraph package. Also, DHHS has a valued tie, where the values range

from 1 (only share information) to 4 (formal collaboration across multiple projects).

As the density suggests, if all values are included then the network is highly int-

erconnected. For this example we will only examine formal collaboration ties (i.e.,

the ‘collab’ edge attribute is 3 or 4). Here we use the igraph subgraph.edges

function to select only those edges. This gives us a new network that is half as dense

as the original.

data(DHHS)

library(intergraph)

iDHHS <- asIgraph(DHHS)

graph.density(iDHHS)

## [1] 0.312

8.2 Social Cohesion 111

iDHHS <- subgraph.edges(iDHHS,E(iDHHS)[collab >2])

graph.density(iDHHS)

## [1] 0.153

To identify the k-core structure in the network, the graph.coreness function

is used. It returns a vector listing the highest core that each vertex belongs to in the

network. The results tell us the k-cores range from 1 to 6.

coreness <- graph.coreness(iDHHS)

table(coreness)

## coreness

##123456

##7625226

maxCoreness <- max(coreness)

maxCoreness

## [1] 6

To better understand the k-core structure, we can plot the network using the

k-core set information. This example illustrates how igraph uses the special vertex

attributes name and color. The name attribute is used by default to label the nodes

in a plot. Here we copy over the vertex names that were stored in the statnet ver-

tex attribute vertex.names. The color vertex attribute is used to set the default

colors of the nodes. Here we add 1 to the k-core values stored in the coreness vector

as a quick and dirty way to pick different colors for each k-core, as well as to avoid

black (Fig. 8.4).

Vname <- get.vertex.attribute(iDHHS,name='vertex.names',

index=V(iDHHS))

V(iDHHS)$name <- Vname

V(iDHHS)$color <- coreness +1

op <- par(mar =rep(0,4))

plot(iDHHS,vertex.label.cex=0.8)

par(op)

To help with the interpretation, we can label the nodes with their k-core member-

ship value. Also, in this example we demonstrate an alternative way to automatically

pick a distinctive set of colors for the nodes (Fig. 8.5).

colors <- rainbow(maxCoreness)

op <- par(mar =rep(0,4))

plot(iDHHS,vertex.label=coreness,

vertex.color=colors[coreness])

par(op)

112 8 Subgroups

ACF-2

AHRQ-1

AHRQ-2

AHRQ-3

AHRQ-4

CDC-1

CDC-2

CDC-3

CDC-4

CDC-5 CDC-6

CDC-7

CDC-8

CDC-9

CDC-10

CDC-11

CDC-12 CMS-1

FDA-1

FDA-2

HRSA-1

HRSA-2

HRSA-3

IHS-2

IHS-1

NIH-1

NIH-2

NIH-3

NIH-4

NIH-5

NIH-6

NIH-7

NIH-8

NIH-9

NIH-10

NIH-11

NIH-12

NIH-13

NIH-14

NIH-16

OGC-1

OGC-2

OS-1

OS-2

OS-3

OS-4

OS-5

SAMHSA-3

Fig. 8.4 DHHS k-core structure

This ﬁgure shows that the center of the network is made up primarily of the

highest k-core. In this case, the 6-core is comprised of 26 of the 54 total nodes.

Because of the nested structure of k-cores, we can further examine the subgroup

patterns by progressively ‘peeling away’ each of the lower k-cores in turn. To do

this we take advantage of the induced.subgraph function (Fig. 8.6).

V(iDHHS)$name <- coreness

V(iDHHS)$color <- colors[coreness]

iDHHS1_6 <- iDHHS

iDHHS2_6 <- induced.subgraph(iDHHS,

vids=which(coreness >1))

iDHHS3_6 <- induced.subgraph(iDHHS,

vids=which(coreness >2))

iDHHS4_6 <- induced.subgraph(iDHHS,

vids=which(coreness >3))

8.2 Social Cohesion 113

Fig. 8.5 DHHS k-core structure, with k-core membership values

iDHHS5_6 <- induced.subgraph(iDHHS,

vids=which(coreness >4))

iDHHS6_6 <- induced.subgraph(iDHHS,

vids=which(coreness >5))

lay <- layout.fruchterman.reingold(iDHHS)

op <- par(mfrow=c(3,2),mar =c(3,0,2,0))

plot(iDHHS1_6,layout=lay,main="All k-cores")

plot(iDHHS2_6,layout=lay[which(coreness >1),],

main="k-cores 2-6")

plot(iDHHS3_6,layout=lay[which(coreness >2),],

main="k-cores 3-6")

plot(iDHHS4_6,layout=lay[which(coreness >3),],

main="k-cores 4-6")

plot(iDHHS5_6,layout=lay[which(coreness >4),],

main="k-cores 5-6")

plot(iDHHS6_6,layout=lay[which(coreness >5),],

main="k-cores 6-6")

par(op)

114 8 Subgroups

All k-cores

k-cores 2-6

k-cores 3-6

k-cores 4-6

k-cores 5-6

k-cores 6-6

Fig. 8.6 Peeling away DHHS k-cores

8.3 Community Detection 115

8.3 Community Detection

Both cliques and k-cores are examples of subgroup identiﬁcation that rely entirely

on the pattern of internal ties deﬁning the particular groups. Network scientists have

developed a wide variety of subgroup identiﬁcation algorithms and heuristics that

deﬁne groups not just based on the internal ties, but also the pattern of ties between

different groups. That is, a subgroup in a network is a set of nodes that has a rel-

atively large number of internal ties, and also relatively few ties from the group to

other parts of the network. These approaches vary in their details, but they are all

designed to identify internally cohesive subgroups that are somewhat separated or

isolated from other groups or nodes. These approaches are sometimes called com-

munity detection algorithms.

8.3.1 Modularity

An important characteristic of a network that is used in many community detection

algorithms is that of modularity. Modularity is a measure of the structure of the net-

work, speciﬁcally the extent to which nodes exhibit clustering where there is greater

density within the clusters and less density between them (Newman 2006). Modu-

larity can be used in an exploratory fashion, where an algorithm tries to maximize

modularity and returns the node classiﬁcation that is found to best explain the ob-

served clustering. Conversely, modularity can be used in a descriptive fashion where

the modularity statistic is calculated for any node classiﬁcation variable of interest.

For example, an analyst can calculate the modularity score for a friendship network

given the gender of network members. Used this way, modularity reﬂects the extent

to which gender explains the observed clustering among the friends in the network.

Modularity is a chance-corrected statistic, and is deﬁned as the fraction of ties

that fall within the given groups minus the expected such fraction if ties were

distributed at random. The modularity statistic can range from −1/2 to +1. The

closer to 1, the more the network exhibits clustering with respect to the given node

grouping.

Consider the following simple example of a network with nine nodes. We have

two categorical vertex attributes which each classify the nodes into three groups

(Fig. 8.7).

g1 <- graph.formula(A-B-C-A,D-E-F-D,G-H-I-G,A-D-G-A)

V(g1)$grp_good <- c(1,1,1,2,2,2,3,3,3)

V(g1)$grp_bad <- c(1,2,3,2,3,1,3,1,2)

op <- par(mfrow=c(1,2))

plot(g1,vertex.color=(V(g1)$grp_good),

vertex.size=20,

116 8 Subgroups

main="Good Grouping")

plot(g1,vertex.color=(V(g1)$grp_bad),

vertex.size=20,

main="Bad Grouping")

par(op)

Good Grouping

Bad Grouping

Fig. 8.7 Modularity example

As the ﬁgure suggests, the clustering that is evident in the network is better

accounted for by the grp good node attribute compared to the grp bad vari-

able. This can be conﬁrmed by calculating the modularity score provided by the

modularity function in igraph.

modularity(g1,V(g1)$grp_good)

## [1] 0.417

modularity(g1,V(g1)$grp_bad)

## [1] -0.333

Real-world social networks are often characterized by clustering, but it is of

course harder to judge the extent of the clustering by eye. Earlier in the chapter

we saw that there was interesting subgroup structure contained in the DHHS net-

work, and that this structure might be partially explained by the DHHS organization

that the person worked for.

library(intergraph)

data(DHHS)

iDHHS <- asIgraph(DHHS)

table(V(iDHHS)$agency)

##012345678910

## 2 4 12 2 2 3 2 16 3 5 3

8.3 Community Detection 117

V(iDHHS)[1:10]$agency

## [1]0011112222

modularity(iDHHS,(V(iDHHS)$agency+1))

## [1] 0.14

The modularity function expects that the node grouping variable is numbered

starting at 1 (and in fact the current version of the function will crash if community

membership has zeros.) In this case, agency is numbered starting at 0 so we add 1

to it before passing it to the modularity function. The modularity score of 0.14

suggests that the DHHS agency does explain some of the clustering that is present

in the network. However, like most network descriptive statistics, the number in and

of itself has little meaning. The interpretation of the network characteristic becomes

more meaningful when it is compared to another relevant measure. For example,

how does the modularity change over time? Or, how does this modularity score

compare to the modularity score for a different vertex attribute on the same network?

As a comparison, we can look at the modularity scores for two other datasets

included in UserNetR.FortheMoreno data, we can see how gender accounts

for subgroup structure. For the Facebook network, we can use the group node

attribute, which designates the type of social group the Facebook friends belong to

(family, work, music, high school, etc.). The results show us that both the Moreno

and Facebook social networks exhibit higher modularity than the DHHS network.

data(Moreno)

iMoreno <- asIgraph(Moreno)

table(V(iMoreno)$gender)

## 1 2

## 16 17

modularity(iMoreno,V(iMoreno)$gender)

## [1] 0.476

data(Facebook)

levels(factor(V(Facebook)$group))

## [1] "B" "C" "F" "G" "H" "M" "S" "W"

grp_num <- as.numeric(factor(V(Facebook)$group))

modularity(Facebook,grp_num)

## [1] 0.615

118 8 Subgroups

8.3.2 Community Detection Algorithms

The main reason that this chapter uses igraph is that it includes support for many

if not most of the existing community detection approaches. Table 8.2 lists the cur-

rently supported algorithms, along with whether each function supports directed

networks, weighted networks (networks with valued ties), and whether the algo-

rithm can be used on networks with more than one component.

Name Function Directed Weighted Components

Edge-betweenness cluster edge betweenness T T T

Leading eigenvector cluster leading eigen F F T

Fast-greedy cluster fast greedy F T T

Louvain cluster louvain F T T

Walktrap cluster walktrap F T F

Label propagation cluster label prop F T F

InfoMAP cluster infomap T T T

Spinglass cluster spinglass F T F

Optimal cluster optimal F T T

Table 8.2 Community detection functions in igraph

The basic workﬂow for conducting community detection in igraph is to run

one of the community detection functions on a network and store the results in

acommunities class object. Then, the identiﬁed subgroups in the network can

be explored using a number of igraph functions that know how to operate with

communities objects. The networks can also be plotted easily to show the results

of the community detection.

For example, consider the simple Moreno friendship network that is clearly div-

ided into two subgroups based on gender. Community detection on this network

proceeds as follows.

cw <- cluster_walktrap(iMoreno)

membership(cw)

## [1]11111111333555513222444

## [24] 2 2 2 2 2 2 2 2 6 6

modularity(cw)

## [1] 0.618

Modularity is fairly high, suggesting that the walktrap algorithm has done a good

job at detecting subgroup structure. The membership function reveals that six dif-

ferent subgroups have been identiﬁed. These are best understood through visualiza-

tion. If the plot function is passed a communities object along with the network it

belongs to, then an attractive plot is produced with each subgroup getting its own

color-coded shaded polygon (Fig. 8.8).

8.3 Community Detection 119

plot(cw, iMoreno)

Once you have a community detection subgroup solution, it can be examined like

any membership vector. For example, you can compare it to an existing partition

based on a node characteristic. In this case, how does a walktrap solution compare

to the speciﬁc agency of the nodes in the DHHS network?

19 20

21 22

27 28

Fig. 8.8 Community detection on Moreno network

cw <- cluster_walktrap(iDHHS)

modularity(cw)

## [1] 0.165

membership(cw)

## [1]45111111111111111111111

##[24]11113333333333333333222

## [47] 2 1 2 1 1 1 1 1

table(V(iDHHS)$agency,membership(cw))

## 12345

## 0 00011

## 1 40000

## 2120000

## 3 20000

## 4 20000

## 5 30000

120 8 Subgroups

## 6 20000

## 7 0 0 16 0 0

## 8 03000

## 9 32000

## 1030000

A common practice is to use more than one community detection algorithm and

compare the results. (Remember that only some algorithms can handle particular

types of networks such as directed networks.) The following examples explore how

different algorithms ﬁnd slightly different subgroups in the Bali terrorism network.

data(Bali)

iBali <- asIgraph(Bali)

cw <- cluster_walktrap(iBali)

modularity(cw)

## [1] 0.283

membership(cw)

## [1]21212212233333222

ceb <- cluster_edge_betweenness(iBali)

modularity(ceb)

## [1] 0.239

membership(ceb)

## [1]11111111122222111

cs <- cluster_spinglass(iBali)

modularity(cs)

## [1] 0.297

membership(cs)

## [1]12131121133333111

cfg <- cluster_fast_greedy(iBali)

modularity(cfg)

## [1] 0.263

membership(cfg)

## [1]22121221133333111

8.3 Community Detection 121

clp <- cluster_label_prop(iBali)

modularity(clp)

## [1] 0.239

membership(clp)

## [1]11111111122222111

cle <- cluster_leading_eigen(iBali)

modularity(cle)

## [1] 0.275

membership(cle)

## [1]11121121122222111

cl <- cluster_louvain(iBali)

modularity(cl)

## [1] 0.297

membership(cl)

## [1]31323313322222333

co <- cluster_optimal(iBali)

modularity(co)

## [1] 0.297

membership(co)

## [1]12131121133333111

These results show that all the detection algorithms identify either two or three

subgroups. Modularity ranges from about 0.24 to 0.30.

The community detection results can be compared to one another using a number

of classiﬁcation comparison metrics, including the adjusted Rand statistic.

table(V(iBali)$role,membership(cw))

## 1 2 3

## BM 0 5 0

## CT 1 2 0

## OA 2 1 0

## SB 0 1 1

## TL 0 0 4

122 8 Subgroups

compare(as.numeric(factor(V(iBali)$role)),cw,

method="adjusted.rand")

## [1] 0.35

compare(cw,ceb,method="adjusted.rand")

## [1] 0.616

compare(cw,cs,method="adjusted.rand")

## [1] 0.89

compare(cw,cfg,method="adjusted.rand")

## [1] 0.669

Finally, we can plot multiple solutions to better understand the similarities and

differences among the different community detection algorithms (Fig. 8.9).

op <- par(mfrow=c(3,2),mar=c(3,0,2,0))

plot(ceb, iBali,vertex.label=V(iBali)$role,

main="Edge Betweenness")

plot(cfg, iBali,vertex.label=V(iBali)$role,

main="Fastgreedy")

plot(clp, iBali,vertex.label=V(iBali)$role,

main="Label Propagation")

plot(cle, iBali,vertex.label=V(iBali)$role,

main="Leading Eigenvector")

plot(cs, iBali,vertex.label=V(iBali)$role,

main="Spinglass")

plot(cw, iBali,vertex.label=V(iBali)$role,

main="Walktrap")

par(op)

8.3 Community Detection 123

Edge Betweenness

Fastgreedy

Label Propagation

BM CT

Leading Eigenvector

Spinglass

TL TL

Walktrap

Fig. 8.9 Community detection comparisons on Bali network

Chapter 9

Afﬁliation Networks

A tribe is a group of people connected to one another, connected

to a leader, and connected to an idea. For millions of years,

human beings have been part of one tribe or another. A group

needs only two things to be a tribe: a shared interest and a way

to communicate. (Seth Godin – Tribes: We Need You to Lead Us)

9.1 Deﬁning Afﬁliation Networks

Until now, all the networks that we have examined are based on direct ties. That

is, the social ties connecting the actors in the social network have been conﬁrmed

through self-report, direct observation, or some other type of data collection that

tells us how actors are directly connected to one another.

However, social scientists are often interested in situations where there may be

the opportunity for social relationships, but these relationships cannot be directly

observed. However, by virtue of occupying the same social situation, we may infer

that there is an opportunity or potential for social connections.

We call this new type of social network an afﬁliation network. An afﬁliation

network is a network where the members are afﬁliated with one another based on

co-membership in a group, or co-participation in some type of event. For example,

students who all belong to the same class can be thought of as being connected to

one another, although we may not know whether they actually have direct social

ties.

The classic example from network science is the case of corporate interlocks.

Company directors have the opportunity to interact with each other when they sit

together on the same corporate board of directors. Moreover, the companies them-

selves can be seen to be connected through their shared director memberships. That

is, when the same director sits on two different company boards, those compa-

nies are connected through that director. Sociologists and political scientists have

used these types of afﬁliation networks to explain how companies tend to behave in

similar ways to one another (Galaskiewicz 1985).

D.A. Luke, A User’s Guide to Network Analysis in R, Use R!,

DOI 10.1007/978-3-319-23883-8 9

125

126 9 Afﬁliation Networks

9.1.1 Afﬁliations as 2-Mode Networks

As a simple example of an afﬁliation network, consider the following data table of

students grouped in classes (Table 9.1).

C1 <- c(1,1,1,0,0,0)

C2 <- c(0,1,1,1,0,0)

C3 <- c(0,0,1,1,1,0)

C4 <- c(0,0,0,0,1,1)

aff.df <- data.frame(C1,C2,C3,C4)

row.names(aff.df) <- c("S1","S2","S3","S4","S5","S6")

C1 C2 C3 C4

S11000

S21100

S31110

S40110

S50011

S60001

Table 9.1 Students grouped by classes

This type of data matrix is called an incidence matrix, and it depicts how nactors

belong to ggroups. In this case we have six students grouped into four classes.

An incidence matrix is similar to an adjacency matrix, but an adjacency matrix is

an nxn square matrix where each dimension refers to the actors in the network. An

incidence matrix, on the other hand, is an nxg rectangular matrix with two different

dimensions: actors and groups. For this reason, afﬁliation networks are also known

as two-mode networks.

9.1.2 Bipartite Graphs

In afﬁliation networks, there are always two types of nodes: one type for the actors,

and another type for the groups or events to which the actors belong. Ties then

connect the actors to those groups. One consequence of this is that there are no direct

ties among actors, and there are no direct ties between the groups. Figure 9.1, which

shows the example student afﬁliation network, illustrates this deﬁning characteristic

of a bipartite graph.

Both statnet and igraph have functionality built in to recognize and operate

on afﬁliation networks. The process with igraph is a little more straightforward,

so it will be used in this chapter.

9.2 Afﬁliation Network Basics 127

library(igraph)

bn <- graph.incidence(aff.df)

plt.x <- c(rep(2,6),rep(4,4))

plt.y <- c(7:2,6:3)

lay <- as.matrix(cbind(plt.x,plt.y))

shapes <- c("circle","square")

colors <- c("blue","red")

plot(bn,vertex.color=colors[V(bn)$type+1],

vertex.shape=shapes[V(bn)$type+1],

vertex.size=10,vertex.label.degree=-pi/2,

vertex.label.dist=1.2,vertex.label.cex=0.9,

layout=lay)

9.2 Afﬁliation Network Basics

9.2.1 Creating Afﬁliation Networks from Incidence Matrices

An afﬁliation network can be stored as an igraph object in a few different ways.

If the underlying data are available as an incidence matrix (e.g., Table 9.1), then this

can be done in one line of code.

bn <- graph.incidence(aff.df)

## IGRAPH UN-B 10 11 --

## + attr: type (v/l), name (v/c)

## + edges (vertex names):

## [1] S1--C1 S2--C1 S2--C2 S3--C1 S3--C2 S3--C3

## [7] S4--C2 S4--C3 S5--C3 S5--C4 S6--C4

The graph.incidence function takes a matrix or data.frame and transforms

it into an afﬁliation network, reading the rows as actors and the columns as the

groups or events. Note that both the row and column names should be deﬁned so

that they can be correctly assigned to the individual actors and groups.

By typing in the name of the igraph object, we get a cryptic two line summary

of the network. The ‘B’ in the ‘UN-B’ string tells us that this is a bipartite net-

work. Furthermore, the second line shows that this network has two vertex attributes:

128 9 Afﬁliation Networks

name stores the name of the vertex, and type is a logical vector that igraph uses

to distinguish between the two different types of nodes (students and classes, in this

case).

Fig. 9.1 Afﬁliation network as bipartite graph

More information about the afﬁliation network can be obtained using traditional

igraph functions.

get.incidence(bn)

## C1 C2 C3 C4

## S1 1 0 0 0

## S2 1 1 0 0

## S3 1 1 1 0

## S4 0 1 1 0

## S5 0 0 1 1

## S6 0 0 0 1

V(bn)$type

## [1] FALSE FALSE FALSE FALSE FALSE FALSE TRUE

## [8] TRUE TRUE TRUE

V(bn)$name

## [1] "S1" "S2" "S3" "S4" "S5" "S6" "C1" "C2" "C3"

## [10] "C4"

9.2 Afﬁliation Network Basics 129

Here we can see the underlying incidence matrix, and the vertex names and types.

The correspondence between the names and the logical type vectors is clear, where

all the students have type =FALSE, and the class nodes have type =TRUE.

9.2.2 Creating Afﬁliation Networks from Edge Lists

For larger networks, it is more common to have the underlying data available as

an edge list. Edge lists can also be translated into an afﬁliation network, as long

nodes of one type (e.g., students) are only connected to nodes of the other type

(e.g., classes). The following code constructs the same example afﬁliation network

from edge list data.

el.df <- data.frame(rbind(c("S1","C1"),

c("S2","C1"),

c("S2","C2"),

c("S3","C1"),

c("S3","C2"),

c("S3","C3"),

c("S4","C2"),

c("S4","C3"),

c("S5","C3"),

c("S5","C4"),

c("S6","C4")))

el.df

## X1 X2

## 1 S1 C1

## 2 S2 C1

## 3 S2 C2

## 4 S3 C1

## 5 S3 C2

## 6 S3 C3

## 7 S4 C2

## 8 S4 C3

## 9 S5 C3

## 10 S5 C4

## 11 S6 C4

bn2 <- graph.data.frame(el.df,directed=FALSE)

bn2

## IGRAPH UN-- 10 11 --

## + attr: name (v/c)

## + edges (vertex names):

130 9 Afﬁliation Networks

## [1] S1--C1 S2--C1 S2--C2 S3--C1 S3--C2 S3--C3

## [7] S4--C2 S4--C3 S5--C3 S5--C4 S6--C4

The above creates an network object, but igraph does not know that it is a

bipartite graph. (Note that the network description lacks the ‘B’ that indicates a

bipartite graph.) To ﬁx this, we can simply set the type vertex attribute. Once this

is done, we have an afﬁliation network object that is formed from a bipartite graph.

V(bn2)$type <- V(bn2)$name %in% el.df[,1]

bn2

## IGRAPH UN-B 10 11 --

## + attr: name (v/c), type (v/l)

## + edges (vertex names):

## [1] S1--C1 S2--C1 S2--C2 S3--C1 S3--C2 S3--C3

## [7] S4--C2 S4--C3 S5--C3 S5--C4 S6--C4

graph.density(bn)==graph.density(bn2)

## [1] TRUE

9.2.3 Plotting Afﬁliation Networks

As with any type of network, afﬁliation networks can be plotted for visual inspec-

tion. However, it is useful to designate different node shapes and colors to make

the afﬁliation structure easier to interpret. Here we will set the students to be blue

circles, and the classes to be red squares. The code shows how the type attribute

can be used as an index into both a shapes and a colors vector to select the appro-

priate shape and color for each node. Note that 1 is added to type index because

as a logical vector it starts at 0, whereas we want to select either the ﬁrst or second

elements of the shapes/colors vectors.

shapes <- c("circle","square")

colors <- c("blue","red")

plot(bn,vertex.color=colors[V(bn)$type+1],

vertex.shape=shapes[V(bn)$type+1],

vertex.size=10,vertex.label.degree=-pi/2,

vertex.label.dist=1.2,vertex.label.cex=0.9)

9.2 Afﬁliation Network Basics 131

Fig. 9.2 Simple plot of afﬁliation network

9.2.4 Projections

By examining Fig. 9.2 we can see how classes are indirectly connected through their

shared students. For example, classes 2 and 3 are indirectly connected through two

shared students (S3 and S4). In comparison, classes 3 and 4 are also indirectly con-

nected, but only via one shared student (S5). We can also focus the examination on

the individual-level nodes. Here the ﬁgure reveals that students 2 and 4 are indirectly

connected by their co-afﬁliation in class 2.

Examining both types of nodes in a two-mode network graphic is often the ﬁrst

step in studying an afﬁliation network. However, it is also useful to examine the

direct connections among the nodes of one type at a time (classes and students, in

this case). This can be done by extracting and visualizing the one-mode projections

of the two-mode afﬁliation network. Every actor by event afﬁliation network can

produce two one-mode networks, one of actors and one of the events or afﬁliations.

In igraph the projections can be obtained again by just one line of code.

The bipartite.projection function returns a list of two igraph network

objects. The ﬁrst network is made up of the direct ties among the ﬁrst mode (in our

case students), and the second network shows the ties among the second mode

(classes).

bn.pr <- bipartite.projection(bn)

bn.pr

## $proj1

## IGRAPH UNW- 6 8 --

## + attr: name (v/c), weight (e/n)

## + edges (vertex names):

132 9 Afﬁliation Networks

## [1] S1--S2 S1--S3 S2--S3 S2--S4 S3--S4 S3--S5

## [7] S4--S5 S5--S6

## $proj2

## IGRAPH UNW- 4 4 --

## + attr: name (v/c), weight (e/n)

## + edges (vertex names):

## [1] C1--C2 C1--C3 C2--C3 C3--C4

Each of the list members can be accessed and treated like a typical igraph

network object, either within the list, or by extracting the list member.

graph.density(bn.pr$proj1)

## [1] 0.533

bn.student <- bn.pr$proj1

bn.class <- bn.pr$proj2

graph.density(bn.student)

## [1] 0.533

The adjacency matrix of each one-mode projection can be obtained with the

get.adjacency function. In the code below, notice how the edge attribute

weight is speciﬁed. This produces a valued adjacency matrix, where the values

indicate how many ties connect any of the nodes. So, for example, the Class adj-

acency matrix indicates that classes 2 and 3 have a weight of 2. This reﬂects the

observation we made earlier that classes 2 and 3 share two students (S3 and S4).

get.adjacency(bn.student,sparse=FALSE,attr="weight")

## S1 S2 S3 S4 S5 S6

##S1011000

##S2102100

##S3120210

##S4012010

##S5001101

##S6000010

get.adjacency(bn.class,sparse=FALSE,attr="weight")

## C1 C2 C3 C4

## C1 0 2 1 0

## C2 2 0 2 0

## C3 1 2 0 1

## C4 0 0 1 0

9.3 Example: Hollywood Actors as an Afﬁliation Network 133

Each of the one-mode projections can, of course, be plotted for visual exami-

nation. Additionally, we can (at least for smaller networks) take advantage of the

weight edge attribute to explore the relative strengths of the ties (Fig. 9.3).

shapes <- c("circle","square")

colors <- c("blue","red")

op <- par(mfrow=c(1,2))

plot(bn.student,vertex.color="blue",

vertex.shape="circle",main="Students",

edge.width=E(bn.student)$weight*2,

vertex.size=15,vertex.label.degree=-pi/2,

vertex.label.dist=1.2,vertex.label.cex=1)

plot(bn.class,vertex.color="red",

vertex.shape="square",main="Classes",

edge.width=E(bn.student)$weight*2,

vertex.size=15,vertex.label.degree=-pi/2,

vertex.label.dist=1.2,vertex.label.cex=1)

par(op)

9.3 Example: Hollywood Actors as an Afﬁliation Network

The data ﬁle hwd in the UseNetR package contains a larger and more interesting

afﬁliation network that can be explored using these techniques. Hollywood actors

are a good example of an afﬁliation network, actors are connected to one another

through the movies in which they appear together. The hwd dataset is an igraph

bipartite graph object. The data are originally from IMDB (www.imdb.com).

The dataset contains the ten most popular movies (as judged by IMBD users) for

Students

Classes

Fig. 9.3 Plots of one-mode projections

134 9 Afﬁliation Networks

each year from 1999 to 2014, and the ﬁrst ten actors listed on each movie’s IMDB

page. In addition to the movie and actor names, each movie has the year of its re-

lease, its IMDB user rating, and the MPAA movie rating (i.e., G, PG, PG-13, and R)

stored as a node characteristic.

9.3.1 Analysis of Entire Hollywood Afﬁliation Network

The ﬁrst steps to analyze these data are to load the ﬁle, and explore the basic afﬁli-

ation structure of the network.

data(hwd)

h1 <- hwd

## IGRAPH UN-B 1365 1600 --

## + attr: name (v/c), type (v/l), year (v/n),

## | IMDBrating (v/n), MPAArating (v/c)

## + edges (vertex names):

## [1] Inception--Leonardo DiCaprio

## [2] Inception--Joseph Gordon-Levitt

## [3] Inception--Ellen Page

## [4] Inception--Tom Hardy

## [5] Inception--Ken Watanabe

## [6] Inception--Dileep Rao

## [7] Inception--Cillian Murphy

## + ... omitted several edges

V(h1)$name[1:10]

## [1] "Inception"

## [2] "Alice in Wonderland"

## [3] "Kick-Ass"

## [4] "Toy Story 3"

## [5] "How to Train Your Dragon"

## [6] "Despicable Me"

## [7] "Scott Pilgrim vs. the World"

## [8] "Hot Tub Time Machine"

## [9] "Harry Potter and the Deathly Hallows: Part 1"

## [10] "Tangled"

V(h1)$type[1:10]

## [1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

## [10] TRUE

V(h1)$IMDBrating[1:10]

9.3 Example: Hollywood Actors as an Afﬁliation Network 135

## [1] 8.8 6.5 7.8 8.4 8.2 7.7 7.5 6.5 7.7 7.9

V(h1)$name[155:165]

## [1] "Notting Hill"

## [2] "Eyes Wide Shut"

## [3] "The Green Mile"

## [4] "10 Things I Hate About You"

## [5] "American Pie"

## [6] "Girl, Interrupted"

## [7] "Leonardo DiCaprio"

## [8] "Joseph Gordon-Levitt"

## [9] "Ellen Page"

## [10] "Tom Hardy"

## [11] "Ken Watanabe"

The summary description of h1 indicates that hwd is indeed a bipartite graph.

We can surmise that the ties link each actor to the movie that actor was in. The

description also reveals that the network has 1,365 nodes and 1,600 ties. This is a

little harder to decipher for an afﬁliation network, but given what we already know

we can ﬁgure out that there are 160 movie nodes, 1,205 actor nodes, and the 1,600

ties arise from each movie having links to just ten actors. (There are only 1,205

actors listed because some actors appear in more than one movie.)

The entire network is too large to show here, but we can examine a small subset

of it before doing more focused analyses. As a ﬁrst step, we can take advantage of

igraph’s ability to store plotting information within the network object itself. In

this case, the node color and shape can be designated by deﬁning these as vertex

attributes. (Compare to how this was done in the previous plotting example, where

the node colors and shapes were designated within the plot function call.)

V(h1)$shape <- ifelse(V(h1)$type==TRUE,

"square","circle")

V(h1)$shape[1:10]

## [1] "square" "square" "square" "square" "square"

## [6] "square" "square" "square" "square" "square"

V(h1)$color <- ifelse(V(h1)$type==TRUE,

"red","lightblue")

For the ﬁrst plot, we will look at a subset of Martin Scorsese movies that were

released in the past 15 years. This example also illustrates how to create a subgraph

by extracting only the edges that are incident to vertices with certain properties

(in this case the name matches one of the three listed Scorsese movies). The key

here is the inc special function of the E() edge iterator. The inc function takes

a vertex sequence as an argument, and returns the incident edges. In this case, we

are extracting all of the edges that are incident to the three Scorsese movies. For

136 9 Afﬁliation Networks

more information see the igraph help entry on iterators, which can be found with

help(E). The resulting graphic highlights the special role of Leonardo DiCaprio

in these Scorsese movies, being the only actor to star in all three (Fig. 9.4).

h2 <- subgraph.edges(h1, E(h1)[inc(V(h1)[name %in%

c("The Wolf of Wall Street","Gangs of New York",

"The Departed")])])

plot(h2, layout = layout_with_kk)

The Wolf of Wall Street

The Departed

Leonardo DiCaprio

Gangs of New York

Jonah Hill

John C. Reilly

Cameron Diaz

Matthew McConaughey

Margot Robbie

Kyle Chandler

Rob Reiner

Jon Favreau

Jean Dujardin

Jon Bernthal

Joanna Lumley

Mark Wahlberg

Jim Broadbent

Kevin Corrigan

Liam Neeson

Anthony Anderson

Matt Damon Jack Nicholson

Martin Sheen

Ray Winstone

Vera Farmiga

Alec Baldwin

Brendan Gleeson

Daniel Day-Lewis

Henry Thomas

Gary Lewis

Stephen Graham

Fig. 9.4 Scorsese afﬁliation network

What can be learned from the entire Hollywood network? Most network descrip-

tive statistics can be applied to afﬁliation networks, but they often need to be adj-

usted either in how they are constructed or how they are interpreted. For example,

the overall density of the afﬁliation network can be easily calculated, but it is not

very meaningful given how the network data were collected (every actor by deﬁni-

tion is connected to a movie) and that there can be no ties among either the movie

nodes or among the actor nodes.

graph.density(h1)

## [1] 0.00172

9.3 Example: Hollywood Actors as an Afﬁliation Network 137

Instead, node degree may be more informative, at least for actors. (In this dataset,

every movie has the same degree =10). The degree function allows speciﬁcation

of which vertices to include, and that is used to select only the actors (for which the

node characteristic type is FALSE).

table(degree(h1,v=V(h1)[type==FALSE]))

##12345678

## 955 165 47 23 11 2 1 1

mean(degree(h1,v=V(h1)[type==FALSE]))

## [1] 1.33

This shows that the vast majority of actors only appeared in one movie, but there

were 15 actors who each starred in ﬁve or more movies since 1999. Across all the

actors, they starred in an average of 1.3 movies. This information can then be used

to identify the busiest actors of the past decade and a half. They owe a lot of thanks

to Harry Potter and Batman!

V(h1)$deg <- degree(h1)

V(h1)[type==FALSE &deg >4]$name

## [1] "Leonardo DiCaprio" "Emma Watson"

## [3] "Richard Griffiths" "Harry Melling"

## [5] "Daniel Radcliffe" "Rupert Grint"

## [7] "James Franco" "Ian McKellen"

## [9] "Martin Freeman" "Bradley Cooper"

## [11] "Christian Bale" "Samuel L. Jackson"

## [13] "Natalie Portman" "Brad Pitt"

## [15] "Liam Neeson"

busy_actor <- data.frame(cbind(

Actor =V(h1)[type==FALSE &deg >4]$name,

Movies =V(h1)[type==FALSE &deg >4]$deg

))

busy_actor[order(busy_actor$Movies,decreasing=TRUE),]

## Actor Movies

## 5 Daniel Radcliffe 8

## 11 Christian Bale 7

## 1 Leonardo DiCaprio 6

## 2 Emma Watson 6

## 3 Richard Griffiths 5

## 4 Harry Melling 5

## 6 Rupert Grint 5

## 7 James Franco 5

138 9 Afﬁliation Networks

## 8 Ian McKellen 5

## 9 Martin Freeman 5

## 10 Bradley Cooper 5

## 12 Samuel L. Jackson 5

## 13 Natalie Portman 5

## 14 Brad Pitt 5

## 15 Liam Neeson 5

This tells us who the busiest actors were. If we wanted to assess the popularity

of the actors based on the popularity of the movies they appeared in, we could do

this by accessing the characteristics of each movie that each actor starred in. This is

slightly more complicated than the previous example, but can be done by utilizing

igraph’s abilities to identify the adjacent neighbors for any node in the graph.

The following code loops through the actor nodes in the network, and sums up

the IMDBrating for all the neighbors of each node. Note that the loop only assigns

the summed IMDBrating scores for the actor nodes (which are listed after the ﬁrst

160 movie nodes).

for (i in 161:1365){

V(h1)[i]$totrating <- sum(V(h1)[nei(i)]$IMDBrating)

}

Once we have this we can once again examine the most popular actors, which is

based on both the number of movies, and the overall popularity of those movies.

max(V(h1)$totrating,na.rm=TRUE)

## [1] 60.9

pop_actor <- data.frame(cbind(

Actor =V(h1)[type==FALSE &totrating >40]$name,

Popularity =V(h1)[type==FALSE &

totrating >40]$totrating))

pop_actor[order(pop_actor$Popularity,decreasing=TRUE),]

## Actor Popularity

## 3 Daniel Radcliffe 60.9

## 4 Christian Bale 55.5

## 1 Leonardo DiCaprio 49.6

## 2 Emma Watson 45

## 5 Brad Pitt 40.5

Finally, network characteristics can always be examined using more traditional

graphical and statistical approaches. For example, we can see if the busiest actors

are starring in more popular movies, on average. First, we calculate an avgrating

characteristic that is based on the mean IMDBrating, rather than the sum. Then,

a simple scatterplot and regression are examined to see the relationship between

9.3 Example: Hollywood Actors as an Afﬁliation Network 139

number of movies and the average ratings of those movies. The results suggest that

there is not a strong relationship between how busy an actor has been and the popu-

larity of their movies. However, the scatterplot also suggests that actors who appear

in less popular movies are most likely to appear in only one or two movies (Fig. 9.5).

for (i in 161:1365){

V(h1)[i]$avgrating <- mean(V(h1)[nei(i)]$IMDBrating)

}

num <- V(h1)[type==FALSE]$deg

avgpop <- V(h1)[type==FALSE]$avgrating

summary(lm(avgpop ˜num))

## Call:

## lm(formula = avgpop ˜ num)

## Residuals:

## Min 1Q Median 3Q Max

## -3.986 -0.433 0.198 0.617 1.614

## Coefficients:

## Estimate Std. Error t value Pr(>|t|)

## (Intercept) 7.3387 0.0544 134.91 <2e-16

## num 0.0471 0.0353 1.34 0.18

## Residual standard error: 0.96 on 1203 df

## Multiple R-sq: 0.00148,Adjusted R-sq: 0.000653

## F-statistic: 1.79 on 1 and 1203 DF, p-value: 0.182

scatter.smooth(num,avgpop,col="lightblue",

ylim=c(2,10),span=.8,

xlab="Number of Movies",

ylab="Avg. Popularity")

9.3.2 Analysis of the Actor and Movie Projections

Following the same procedures as presented in Sect. 9.2.4, the two projections of

the Hollywood afﬁliation network can be created and analyzed. This will produce

an actor network where actors have ties if they starred together in the same movie,

and a movie network where the movies are connected if they shared the same actors.

The actor projection will thus have 1,205 nodes, and the movie projection network

will have 160 nodes.

140 9 Afﬁliation Networks

12345678

108642

Number of Movies

Avg. Popularity

Fig. 9.5 Relationship between actor activity and popularity

h1.pr <- bipartite.projection(h1)

h1.act <- h1.pr$proj1

h1.mov <- h1.pr$proj2

h1.act

## IGRAPH UNW- 1205 6903 --

## + attr: name (v/c), year (v/n), IMDBrating

## | (v/n), MPAArating (v/c), shape (v/c),

## | color (v/c), deg (v/n), totrating (v/n),

## | avgrating (v/n), weight (e/n)

## + edges (vertex names):

## [1] Leonardo DiCaprio--Joseph Gordon-Levitt

## [2] Leonardo DiCaprio--Ellen Page

## [3] Leonardo DiCaprio--Tom Hardy

## [4] Leonardo DiCaprio--Ken Watanabe

## [5] Leonardo DiCaprio--Dileep Rao

## + ... omitted several edges

h1.mov

## IGRAPH UNW- 160 472 --

## + attr: name (v/c), year (v/n), IMDBrating

## | (v/n), MPAArating (v/c), shape (v/c),

## | color (v/c), deg (v/n), totrating (v/n),

## | avgrating (v/n), weight (e/n)

## + edges (vertex names):

9.3 Example: Hollywood Actors as an Afﬁliation Network 141

## [1] Inception--The Wolf of Wall Street

## [2] Inception--Django Unchained

## [3] Inception--The Departed

## [4] Inception--Gangs of New York

## [5] Inception--Catch Me If You Can

## + ... omitted several edges

Fig. 9.6 Movie afﬁliation network

In this ﬁgure, the entire movie network is presented, with node size based on the

IMDBrating, so that more popular movies have larger nodes (Fig. 9.6).

op <- par(mar =rep(0,4))

plot(h1.mov,vertex.color="red",

vertex.shape="circle",

vertex.size=(V(h1.mov)$IMDBrating)-3,

vertex.label=NA)

142 9 Afﬁliation Networks

par(op)

Some basic network descriptives provide more information about the Hollywood

movie network. Although there are some isolated movies (i.e., movies that did not

share actors with any of the other movies), most (148) of the movies form a large

connected component.

graph.density(h1.mov)

## [1] 0.0371

no.clusters(h1.mov)

## [1] 12

clusters(h1.mov)$csize

##[1]1481111112111

## [12] 1

table(E(h1.mov)$weight)

##123456710

##4112112166123

The complete movie network can be ﬁltered to examine the single large con-

nected component. In the next ﬁgure the edge width has been set to equal the square

root of weight edge attribute. This results in the ties being thicker for movies that

share more actors between them (Fig. 9.7).

h2.mov <- induced.subgraph(h1.mov,

vids=clusters(h1.mov)$membership==1)

plot(h2.mov,vertex.color="red",

edge.width=sqrt(E(h1.mov)$weight),

vertex.shape="circle",

vertex.size=(V(h2.mov)$IMDBrating)-3,

vertex.label=NA)

The previous ﬁgure is still large and the relatively high density makes it some-

what challenging to interpret any interesting structural features. To help with that,

we can identify the higher density cores of the graph, and use that to ‘zoom in’ on

the more interconnected part of the network. (See Chap. 8for more information.)

This network is small enough that we can add node labels to help with the inter-

pretation. This helps us see that the most tightly connected sections of the network

9.3 Example: Hollywood Actors as an Afﬁliation Network 143

Fig. 9.7 Largest component of movie afﬁliation network

correspond to popular movie series, in particular Harry Potter, Batman, Star Wars,

and The Hobbit. This makes sense because movies in a series will naturally share

many or most of the same actors (Fig. 9.8).

table(graph.coreness(h2.mov))

##1234567

## 11 5 23 65 29 7 8

h3.mov <- induced.subgraph(h2.mov,

vids=graph.coreness(h2.mov)>4)

h3.mov

## IGRAPH UNW- 44 158 --

## + attr: name (v/c), year (v/n), IMDBrating

## | (v/n), MPAArating (v/c), shape (v/c),

## | color (v/c), deg (v/n), totrating (v/n),

144 9 Afﬁliation Networks

## | avgrating (v/n), weight (e/n)

## + edges (vertex names):

## [1] Inception--The Wolf of Wall Street

## [2] Inception--Django Unchained

## [3] Inception--The Dark Knight Rises

## [4] Inception--The Dark Knight

## [5] Inception--The Departed

## + ... omitted several edges

plot(h3.mov,vertex.color="red",

vertex.shape="circle",

edge.width=sqrt(E(h1.mov)$weight),

vertex.label.cex=0.7,vertex.label.color="darkgreen",

vertex.label.dist=0.3,

vertex.size=(V(h3.mov)$IMDBrating)-3)

Inception

Alice in Wonderland

Hot Tub Time Machine

Harry Potter and the Deathly Hallows: Part 1

The Interview

The Hobbit: The Battle of the Five Armies

Exodus: Gods and Kings

The Hobbit: The Desolation of Smaug

The Wolf of Wall Street

American Hustle

This Is the End

The Hobbit: An Unexpected Journey

Django Unchained

The Dark Knight Rises

Harry Potter and the Deathly Hallows: Part 2

Arthur Christmas

Captain America: The First Avenger

Rise of the Planet of the Apes

Harry Potter and the Half-Blood Prince

Nativity!

The Dark Knight

Pineapple Express

Taken

Superbad

Harry Potter and the Order of the Phoenix

Hot Fuzz

The Prestige

The Departed

Harry Potter and the Goblet of Fire

Batman Begins

Star Wars: Episode III - Revenge of the Sith

V for Vendetta

Harry Potter and the Prisoner of Azkaban

Love Actually

Big Fish

Harry Potter and the Chamber of Secrets

Gangs of New York

Spider-Man

Catch Me If You Can

Star Wars: Episode II - Attack of the Clones

Harry Potter and the Sorcerer's Stone

American Psycho

X-Men

Star Wars: Episode I - The Phantom Menace

Fig. 9.8 Core movie afﬁliation network

Part IV

Modeling

Chapter 10

Random Network Models

What is this? A center for ants? How can we be expected to

teachchildren tolearn how to read...iftheycan’teven ﬁtinside

the building? (Derek Zoolander,inthemovieZoolander,after

looking at a miniature model of a school building.)

10.1 The Role of Network Models

According to Linton Freeman (2004), modern social network analysis has four main

characteristics:

1. It is motivated by a structural intuition based on ties linking social actors;

2. It is grounded in systematic empirical data;

3. It draws heavily on graphic imagery; and

4. It relies on the use of mathematical and/or computational models.

The preceding sections of this book focused on the ﬁrst three elements of

Freeman’s characterization. The next four chapters now turn to consider his last

point, the utility of modeling in network analysis. Scientiﬁc models are simpliﬁed

descriptions of the real world that are used to predict or explain the characteristics

or behavior of the phenomenon of interest. Models can be used in network science

in the same way. With network models we can move beyond simple description to

build and test hypotheses about network structures, formation processes, and net-

work dynamics

In this chapter, a number of basic mathematical models of network structure

and formation are covered. These are important models in the history of network

science, but they are still useful today to provide insight into fundamental properties

of social networks, to serve as baseline or comparison models for empirical social

networks, and to act as building blocks for more complex network simulations.

Well over a dozen functions are provided in igraph for generating random net-

works based on a number of mathematical algorithms and heuristics. These all use

‘game’ as the ﬁnal part of the function name, for example barabasi.game()

produces scale-free random graphs based on the Baraba´

si-Albert model (1999). In

the rest of this chapter, a number of important mathematical network models avail-

able in igraph are presented, along with some examples of how to use these mod-

els to explore network properties and as comparisons to observed, empirical social

networks.

D.A. Luke, A User’s Guide to Network Analysis in R, Use R!,

DOI 10.1007/978-3-319-23883-8 10

147

148 10 Random Network Models

10.2 Models of Network Structure and Formation

10.2.1 Erd˝

os-R´

enyi Random Graph Model

The earliest historically, and still one of the most important mathematical models

of network structure, is the random graph model ﬁrst developed by Paul Erd˝

os and

Alfred R´

enyi in the late 1950s and early 1960s (Newman 2010). This is sometimes

called the Poisson random graph model (because of the Poisson degree distribu-

tion of large random graphs), or sometimes even just the random graph model.The

model is quite simple, G(n,m), where a random graph Gis deﬁned with nvertices

and medges among those vertices chosen randomly. An equivalent model that is

easier to work with is G(n,p), where instead of specifying medges, each edge app-

ears in the graph with probability p. This random graph model is implemented in

igraph with the erdos.reny.game() function. A random graph is produced

by specifying the size of the desired network, and either the number of edges, or

the probability of observing an edge. The type argument is used to specify whether

the second argument should be interpreted as probability of an edge p, or number of

edges m.

library(igraph)

g<- erdos.renyi.game(n=12,10,type='gnm')

## IGRAPH U--- 12 10 -- Erdos renyi (gnm) graph

## + attr: name (g/c), type (g/c), loops (g/l),

## | m (g/n)

## + edges:

## [1] 4-- 5 3-- 6 2-- 8 1-- 9 8-- 9 8--10 1--11

## [8] 6--11 8--12 9--12

graph.density(g)

## [1] 0.152

The random nature of the graphs can be seen by producing and examining multi-

ple graphs. In each case the number of vertices is the same, but the ties are randomly

determined (Fig. 10.1).

op <- par(mar=c(0,1,3,1),mfrow=c(1,2))

plot(erdos.renyi.game(n=12,10,type='gnm'),

vertex.color=2,

main="First random graph")

plot(erdos.renyi.game(n=12,10,type='gnm'),

vertex.color=4,

main="Second random graph")

par(op)

10.2 Models of Network Structure and Formation 149

First random graph

Second random graph

Fig. 10.1 Two random graphs

Despite the simplicity of the random graph model, it has led to a number of

important discoveries about network structures. First, as suggested above, for large

nthe network will have a Poisson degree distribution (Fig. 10.2).

g<- erdos.renyi.game(n=1000,.005,type='gnp')

plot(degree.distribution(g),

type="b",xlab="Degree",ylab="Proportion")

More unexpectedly, it turns out that random graphs become entirely connected

for fairly low values of average degree. That means even when edges are determined

randomly, each individual network member does not have to be connected to too

many other members for the network itself to be connected (i.e., the network has

only one component). More precisely, if pis greater than ln

n, then the random graph

is likely to be connected in one large component (Newman 2010). The average

degree of a random graph, c, is related to graph size and edge probability:

c=(n−1)p

So this means that across the range of network sizes typically seen in social net-

work analysis (say, 100–10,000), the average degree required to have a completely

connected network will be less than approximately 12. The following random graph

simulation and plot demonstrates this relationship (Fig. 10.3).

crnd <- runif(500,1,8)

cmp_prp <- sapply(crnd,function(x)

max(clusters(erdos.renyi.game(n=1000,

p=x/999))$csize)/1000)

150 10 Random Network Models

51015

0.00 0.05 0.10 0.15

Degree

Proportion

Fig. 10.2 Degree distribution for G with n = 1,000 and p= 0.005

smoothingSpline <- smooth.spline(crnd,cmp_prp,

spar=0.25)

plot(crnd,cmp_prp,col='grey60',

xlab="Avg. Degree",

ylab="Largest Component Proportion")

lines(smoothingSpline,lwd=1.5)

This demonstration requires some unpacking to easily understand. First, a vector

is created with 500 random values ranging from one to eight. This will be used as

input to a function that will create 500 random graphs, with average degree varying

from 1 to 8. Next, the sapply() function is used to call the random graph func-

tion repeatedly, and assign the results to cmp prp. The function itself creates each

random graph with 1,000 nodes, and with pequal to the desired average degree div-

ided by 999. (From above, p=c

n−1.) Once the random graph is produced, the size

of the largest component is calculated using the clusters() function. This num-

ber is then divided by the size of the network (1,000) to get the proportion of the

network accounted for by the largest component. The results show that for random

networks of 1,000 nodes, the network will be almost or completely connected when

the average degree is larger than four or ﬁve.

Another surprising property of random graphs is that the connected random

graphs are quite compact. That is, the diameter of the largest components in ran-

dom graphs stays relatively small even for large networks.

n_vect <- rep(c(50,100,500,1000,5000),each=50)

g_diam <- sapply(n_vect,function(x)

10.2 Models of Network Structure and Formation 151

12345678

0.2 0.4 0.6 0.8 1.0

Avg. Degree

Largest Component Proportion

Fig. 10.3 Relationship of average degree and connectedness in random graphs

diameter(erdos.renyi.game(n=x,p=6/(x-1))))

library(lattice)

bwplot(g_diam ˜factor(n_vect), panel = panel.violin,

xlab ="Network Size",ylab ="Diameter")

The above code runs a total of 250 simulations, producing random graphs from

50 to 5,000 nodes. As the plot shows, although the size of the graphs increases

across two orders of magnitude, the diameter of the largest component in each graph

increases much more slowly, from about ﬁve to ten (Fig. 10.4). These two character-

istics of random graphs: being completely connected with low average degree, and

the diameter increasing slowly relative to graph size, may be partly responsible for

some of the ‘small-world’ characteristics of real-world social networks (Newman

2010).

10.2.2 Small-World Model

The Erd˝

os-R´

enyi random graph model has one major limitation in that it does not

describe the properties of many real-world social networks. In particular, fully ran-

dom graphs have degree distributions that do not match observed networks very

well, and they also have quite low levels of clustering (transitivity).

One type of model, called the small-world model by Watts and Strogatz (1998),

produces random networks that are somewhat more realistic than Erd˝

os-R´

enyi

152 10 Random Network Models

graphs. In particular, small-world model networks have more realistic levels of tran-

sitivity along with small diameters.

The small-world model starts with a circle of nodes, where each node is con-

nected to its cimmediate neighbors (forming a formal lattice structure). Then, a

small number of existing edges are rewired, where they are removed and then rep-

laced with another tie that connects two random nodes. If the rewiring probability

is 0, then we end up with the original lattice network. When pis 1, then we have an

Erd˝

os-R´

enyi random graph. The main interesting discovery of Watts and Strogatz

(and others), is that only a small fraction of ties needs to be rewired to dramatically

reduce the diameter of the network.

Network Size

Diameter

50 100 500 1000 5000

Fig. 10.4 Relationship of random graph size and diameter, for average degree =6

Figure 10.5 shows how various small-world model networks look with different

rewiring probabilities. The watts.strogatz.game() is called to produce a

small-world network of 30 nodes. Setting the option nei=2 (for neighborhood)

will start the network with each node tied to the closest two neighbors on either

side. This results in each node having degree =4.

g1 <- watts.strogatz.game(dim=1,size=30,nei=2,p=0)

g2 <- watts.strogatz.game(dim=1,size=30,nei=2,p=.05)

g3 <- watts.strogatz.game(dim=1,size=30,nei=2,p=.20)

g4 <- watts.strogatz.game(dim=1,size=30,nei=2,p=1)

op <- par(mar=c(2,1,3,1),mfrow=c(2,2))

plot(g1,vertex.label=NA,layout=layout_with_kk,

10.2 Models of Network Structure and Formation 153

main=expression(paste(italic(p),"=0")))

plot(g2,vertex.label=NA,

main=expression(paste(italic(p)," = .05")))

plot(g3,vertex.label=NA,

main=expression(paste(italic(p)," = .20")))

plot(g4,vertex.label=NA,

main=expression(paste(italic(p),"=1")))

par(op)

p = 0 p = .05

p = .20 p = 1

Fig. 10.5 Small-world models with increasing rewiring probabilities

The following simulation and ﬁgure shows how quickly rewiring reduces the

diameter of a network in the small-world model. Working with a network with 100

nodes, each node starts out connected to its two neighbors on each side. The graph

will thus have 200 edges. The starting diameter of the lattice network is 25 (getting

from one node to the other side of the circle takes 25 steps).

154 10 Random Network Models

g100 <- watts.strogatz.game(dim=1,size=100,nei=2,p=0)

g100

## IGRAPH U--- 100 200 -- Watts-Strogatz random grap

## + attr: name (g/c), dim (g/n), size (g/n),

## | nei (g/n), p (g/n), loops (g/l), multiple

## | (g/l)

## + edges:

## [1] 1-- 2 2-- 3 3-- 4 4-- 5 5-- 6 6-- 7

## [7] 7-- 8 8-- 9 9--10 10--11 11--12 12--13

## [13] 13--14 14--15 15--16 16--17 17--18 18--19

## [19] 19--20 20--21 21--22 22--23 23--24 24--25

## [25] 25--26 26--27 27--28 28--29 29--30 30--31

## [31] 31--32 32--33 33--34 34--35 35--36 36--37

## + ... omitted several edges

diameter(g100)

## [1] 25

The simulation is set to calculate 300 networks, ten each for the number of edges

to rewire ranging from 1 to 30. Because we know how many edges are in each graph

(200), the rewiring probability can be calculated by the number of rewired edges

divided by total number of edges. If 30 edges are rewired, then, the probability is

0.15.

p_vect <- rep(1:30,each=10)

g_diam <- sapply(p_vect,function(x)

diameter(watts.strogatz.game(dim=1,size=100,

nei=2,p=x/200)))

smoothingSpline =smooth.spline(p_vect, g_diam,

spar=0.35)

plot(jitter(p_vect,1),g_diam,col='grey60',

xlab="Number of Rewired Edges",

ylab="Diameter")

lines(smoothingSpline,lwd=1.5)

The plot demonstrates that after only rewiring ten of the edges (p=0.05), the

diameter has shrunk at least 60 %, from 25 to about 10 (Fig. 10.6).

10.2.3 Scale-Free Models

An important limitation of the previous two mathematical network models is that

they produce graphs with degree distributions that are not representative of many

10.2 Models of Network Structure and Formation 155

302520151050

10 15 20 25

Number of Rewired Edges

Diameter

Fig. 10.6 Relationship of rewiring probability to network diameter for the small-world model

real-world social networks. Numerous studies, in fact, have shown that a wide vari-

ety of observed networks have heavy-tailed degree distributions that approximately

follow a power law. These are typically called scale-free networks. For example,

both the network of sexual partners and the World-Wide-Web exhibit this scale-free

pattern (Broder et al. 2000; Liljeros et al. 2001). That is, some people have many

sexual partners (high degree), but most people have a small number of sexual part-

ners. Similarly, some websites have a very large number of other websites connected

to them, but most websites have only a few connections.

How does this power-law characteristic feature of scale-free social networks

arise? A number of network scientists have explored this question, and have det-

ermined that a network formation process of cumulative advantage,orpreferential

attachment can explain this. That is, as networks grow, new nodes are more likely to

form ties with other nodes that already have many ties, due to their visibility in the

network. This ‘rich-gets-richer’ phenomena has been shown to lead to the power-

law distribution in networks (de Solla Price 1976; Barab´

asi and Albert 1999).

The preferential attachment model of Barab´

asi and Albert is implemented in

igraph with the barabasi.game() function. This is a more complicated alg-

orithm than those for the previous models, partly because this is a network growth

model, not just a static network structure model.

Figure 10.7 displays a 500-node network that is formed with this preferential

attachment model. The default behavior of the algorithm is that as each new node is

added to the network, it is connected to another node in the network, with probability

proportional to the degree of that node. Thus, some nodes in the network will end

up with many more ties than most of the other nodes. The code for this example

156 10 Random Network Models

highlights these hubs by coloring the nodes with degree >9 red. Also, the nodes

are sized based on their degree, using the rescale() function from Chap. 5.

g<- barabasi.game(500,directed =FALSE)

V(g)$color <- "lightblue"

V(g)[degree(g) >9]$color <- "red"

node_size <- rescale(node_char =degree(g), low =2,

high =8)

plot(g, vertex.label =NA,vertex.size = node_size)

We can see the heavy-tail distribution in a few ways. The median degree is 1,

and the mean is close to 2. The highest degree is 27. The pattern is easily seen in

Fig. 10.8. The left panel shows the raw degree distribution, while the right panel

displays the same degree distribution, but on log-scales. If the distribution follows a

power-law, then the datapoints should fall along a straight line (at least for the tail).

(Note that it is difﬁcult to assess power-law relationships with small networks.)

median(degree(g))

## [1] 1

mean(degree(g))

## [1] 2

Fig. 10.7 Example scale-free network with default options

10.2 Models of Network Structure and Formation 157

table(degree(g))

##1234567910111214

##3148641231110313221

## 16 19 27

##111

op <- par(mfrow=c(1,2))

plot(degree.distribution(g),xlab="Degree",

ylab="Proportion")

plot(degree.distribution(g),log='xy',

xlab="Degree",ylab="Proportion")

par(op)

The user can adjust a number of parameters in the barabasi.game() func-

tion to produce a wide variety of preferential attachment networks. For example, the

following code produces a network that might be viewed as a little more realistic

than that shown in the previous ﬁgure. Here, instead of each new node connecting to

exactly one other node, the out.dist option is used to specify a distribution of tie

probabilities. In this case a new node will have a tie to 0 other nodes (isolate) 25 %

of the time, will be tied to one other node 50 % of the time, and to two nodes 25 % of

0 5 10 15 20 25

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Degree

Proportion

12 5 1020

0.002 0.005 0.010 0.020 0.050 0.100 0.200 0.500

Degree

Proportion

Fig. 10.8 Degree distribution of scale-free model (linear and log-linear scales)

158 10 Random Network Models

the time. Also, the zero.appeal option is used to make it somewhat more likely

that new nodes are connected to previously existing isolates. Figure 10.9 shows that

the resulting graph has produced a number of isolates and slightly fewer nodes with

high degree.

g<- barabasi.game(500,out.dist =c(0.25,0.5,0.25),

directed =FALSE,zero.appeal =1)

V(g)$color <- "lightblue"

V(g)[degree(g) >9]$color <- "red"

node_size <- rescale(node_char =degree(g), low =2,

high =8)

plot(g, vertex.label =NA,vertex.size = node_size)

Fig. 10.9 Example scale-free network with modiﬁed options

Finally, to illustrate how preferential attachment networks grow, the following

ﬁgure shows what the networks look like at four different stages, with graph sizes

of 10, 25, 50, and 100 nodes (Fig. 10.10).

g1 <- barabasi.game(10,m=1,directed=FALSE)

g2 <- barabasi.game(25,m=1,directed=FALSE)

g3 <- barabasi.game(50,m=1,directed=FALSE)

10.2 Models of Network Structure and Formation 159

g4 <- barabasi.game(100,m=1,directed=FALSE)

op <- par(mfrow=c(2,2),mar=c(4,0,1,0))

plot(g1, vertex.label=NA,vertex.size =3,

xlab ="n = 10")

plot(g2, vertex.label=NA,vertex.size =3,

xlab ="n = 25")

plot(g3, vertex.label=NA,vertex.size =3,

xlab ="n = 50")

plot(g4, vertex.label=NA,vertex.size =3,

xlab ="n = 100")

par(op)

n = 10 n = 25

n = 50 n = 100

Fig. 10.10 Growth of networks using preferential attachment model

160 10 Random Network Models

10.3 Comparing Random Models to Empirical Networks

The mathematical models described here, as well as many others, have been used

to study the theoretical properties of networks. These models can also be useful as

comparisons to empirical social networks. As as simple example of this, we can

explore some basic network properties of the lhds data taken from Harris’ 2013

book on exponential random graph models (see Chap. 11). The lhds network is

made up of communication ties among 1,283 leaders of local public health depart-

ments. The network has quite low density, although the average degree is over four

(Fig. 10.11).

data(lhds)

ilhds <- asIgraph(lhds)

ilhds

## IGRAPH U--- 1283 2708 --

## + attr: title (g/c), hivscreen (v/c), na

## | (v/l), nutrition (v/c), popmil (v/n),

## | state (v/c), vertex.names (v/c), years

## | (v/n), na (e/l)

## + edges:

## [1] 2-- 10 2-- 11 2-- 19 2-- 20 5--1003

## [6] 5-- 6 6-- 11 6-- 17 10-- 11 11-- 19

## [11] 11-- 26 2-- 12 6-- 12 10-- 12 11-- 12

## [16] 12-- 19 12-- 26 9-- 14 14-- 15 14-- 18

## [21] 14-- 25 14-- 27 14-- 226 14-- 414 14-- 697

## + ... omitted several edges

graph.density(ilhds)

## [1] 0.00329

mean(degree(ilhds))

## [1] 4.22

The following code builds three network models that have the same size and

approximately the same density as the lhds network. By comparing the charac-

teristics of the network models with the empirical network, we can highlight the

interesting or important characteristics of the empirical network that might be worth

further exploration (by using, for example, the types of statistical modeling and sim-

ulation approaches presented in the next few chapters).

g_rnd <- erdos.renyi.game(1283,.0033,type='gnp')

g_smwrld <- watts.strogatz.game(dim=1,size=1283,

nei=2,p=.25)

10.3 Comparing Random Models to Empirical Networks 161

g_prfatt <- barabasi.game(1283,out.dist=c(.15,.6,.25),

directed=FALSE,zero.appeal=2)

Table 10.1 presents some descriptive network statistics for the three models and

the lhds network. Although each model captures some of the characteristics of

the observed network, none of them match across all the statistics. In particular, the

lhds network has much higher transitivity than any of the models (Fig. 10.12).

Fig. 10.11 Local health department communication network

Name Size Density Avg. degree Transitivity Isolates

Erdos-Renyi 1283 0.003 4.404 0.002 21

Small world 1283 0.003 4.000 0.088 1

Preferential attachment 1283 0.002 2.195 0.003 109

Health department 1283 0.003 4.221 0.306 58

Table 10.1 Comparison of model and empirical network characteristics

162 10 Random Network Models

0 5 10 15 20 25

0.00 0.05 0.10 0.15 0.20

0.00 0.10 0.20 0.30

Erdos-Renyi Random Graph

Degree

0 5 10 15 20 25

Degree

Small World

0 5 10 15 20 25

0.30 0.1 0.2 0.3

0.00 0.05 0.10 0.15

Preferential Attachment

Degree

0 5 10 15 20 25

Degree

Local Health Dept. Network

Fig. 10.12 Comparison of degree distributions for random models and empirical local health de-

partment network

Chapter 11

Statistical Network Models

Prediction and explanation are exactly symmetrical.

Explanations are, in effect, predictions about what has

happened; predictions are explanations about what’s going to

happen. (John Rogers Searle)

11.1 Introduction

As suggested in the previous chapter, for most of the history of network science

analysts were limited to network visualization and network description. Only in

the past couple of decades has network statistical theory and computational power

developed enough to allow for valid and feasible statistical modeling of networks.

The primary barrier to statistical network modeling was the fundamental assumption

of independence of observations that underlies much of traditional statistical theory.

Networks by deﬁnition are non-independent. If you know that one actor is tied to

another actor, you have information about the second actor that is dependent on the

ﬁrst.

Over the years statistical theorists gradually developed increasingly sophisticated

models that could be applied to empirical network data, including dyadic depen-

dence and dyadic independence models (such as p*,seeHarris2013). The focus of

this chapter is on exponential random graph models (ERGMs), which have turned

out to be the most powerful, ﬂexible, and widely used modeling approach for build-

ing and testing statistical models of networks.

An ERGM is a true generative statistical model of network structure and char-

acteristics (Hunter et al. 2008). This means that inferential hypotheses can be pro-

posed and tested. It is generative in the sense that characteristics of the individual

elements in the network (i.e., actors) and local structural properties can be used to

predict properties of the entire network (e.g., diameter, degree distribution, etc.).

ERGMs are popular for at least four reasons. First, they can handle the complex

dependencies of network data without the types of degeneracy problems that were

frequently encountered in earlier network models. Second, ERGMs are ﬂexible and

can handle many different types of predictors and covariates. Third, the generative

approach where overall network characteristics are predicted from individual actor

and local structural properties enhances the validity of the models. Finally, ERGM

D.A. Luke, A User’s Guide to Network Analysis in R, Use R!,

DOI 10.1007/978-3-319-23883-8 11

163

164 11 Statistical Network Models

models have been implemented in programming suites and statistical packages such

as R, making it easier for applied analysts to build, test, and disseminate the results

of their network models.

ERGMs are ﬁt using Monte Carlo Markov Chain maximum-likelihood estima-

tion (MCMC). The estimates of the statistical parameters are based on an underlying

simulation, where many (typically thousands) of networks are produced to reﬂect

the particular model being tested. One way to express the basic ERGM model is:

P(yij =1|YC

ij)=1

cexp{

∑

k=1

kzk(y)}

This shows that the model is predicting the probability of a tie between actors i

and j, conditional on the rest of the network (all other ties). The thetas (

k)are the

coefﬁcients of the network statistics of interest, one for each of the Kincluded statis-

tics, zk(y).1

cis simply a normalizing constant that ensures that the probabilities

stay within 0 and 1. (See Harris 2013, for more details.)

ERGMs are implemented in the ergm package that is contained in the statnet

suite of network analysis packages in R. It is actively maintained and developed by

network scientists at the University of Washington, among others. More technical

details of ergm are provided in excellent papers by Hunter and colleagues (2008)

as well as Goodreau (2007).

As suggested above, one of the strengths of ERGMs is the ability to handle a wide

variety of predictors. In fact, the documentation for ergm lists more than a hundred

possible types of terms that can be included in an ergm model speciﬁcation. It is

easier to navigate the possibilities once one knows that all the possible predictors

in an ERGM fall into one of four broad categories: node-level predictors, dyadic

predictors, relational predictors, and local structural predictors.

Table 11.1 lists these categories, along with some of the most commonly used

ergm terms for each type. The ﬁrst type of predictor is node or actor-level charac-

teristics, where having a particular characteristic is hypothesized to affect the like-

lihood of observing a tie. For example, if you hypothesize that girls are more likely

to make friends than boys in middle school, then you could use actor gender as

a node-level predictor. Dyad-level predictors are used when you hypothesize that

the characteristics of both actors in a dyad may inﬂuence the probability of obs-

erving a tie between those two actors. These types of predictors allow you to test

hypotheses of assortative or disassortative mixing in networks, leading to patterns

of homophily or heterophily. For example, if you think that friendships are more

likely to be formed within the same grade in a middle school (assortative mixing),

then you could use grade as a dyad-level predictor. The third type of predictor in

ERGMs is a powerful option to use information about other relationships or ties

when predicting the observed ties in a network. That is, you can use one type of

network tie to predict a second type of tie (as long as they are both collected on the

same set of network members). Finally, information about local structural properties

of the observed network can be used as model covariates. This, for example, allows

the network model to be conditioned on the observed degree distribution, or on the

level of transitivity (closed triangles) that is observed.

11.2 Building Exponential Random Graph Models 165

Predictor type Term

Node nodefactor

nodecov

Dyad nodemix

nodematch

absdiff

Relation edgecov

Structure gwdegree

gwdsp

gwesp

Table 11.1 Common ERGM terms

In the following examples, we will build a series of ERGMs that use predictors

from each of these four broad types of ergm terms. For more detailed information

about the terms included in ergm,seehelp(’ergm-terms’). A more general

overview is provided by Morris, Handcock, and Hunter (2008). Finally, ergm inc-

ludes a helpful vignette that shows how the various terms are related to one another

(vignette(’ergm-term-crossRef’)).

11.2 Building Exponential Random Graph Models

To explore a number of the stochastic modeling possibilities of the ergm package,

we will use network data that describe interorganizational relationships among 25

agencies within the Indiana state tobacco control program in 2010. These data inc-

lude three different types of interorganizational ties: frequency of contact, level of

collaboration, and whether each pair of agencies communicated with one another

about a particular evidence-based guideline published by the Center’s for Disease

Control and Prevention (CDC), called Best Practices for Tobacco Control.This

latter relationship is conceptualized as a type of dissemination tie. The following

example models will focus on predicting the pattern of these dissemination ties

among the Indiana tobacco control organizations.

The data are included in the UserNetR package as a network list object called

TCnetworks. The network data include a number of node characteristics (e.g.,

tob yrs, which records how long an agency has been working in tobacco control),

edge characteristics, and a sociomatrix (TCdist) which contains the geographic

distance between each pair of agencies. For more information, see the help ﬁle for

TCnetworks.

Prior to modeling, the network data need to be extracted from the list object, and

any preliminary descriptive and visualization tasks can be conducted.

data(TCnetworks)

TCcnt <- TCnetworks$TCcnt

TCcoll <- TCnetworks$TCcoll

166 11 Statistical Network Models

TCdiss <- TCnetworks$TCdiss

TCdist <- TCnetworks$TCdist

summary(TCdiss,print.adj=FALSE)

## Network attributes:

## vertices = 25

## directed = FALSE

## hyper = FALSE

## loops = FALSE

## multiple = FALSE

## bipartite = FALSE

## title = IN_Diffusion

## total edges = 103

## missing edges = 0

## non-missing edges = 103

## density = 0.343

## Vertex attributes:

## agency_cat:

## numeric valued attribute

## attribute summary:

## Min. 1st Qu. Median Mean 3rd Qu. Max.

## 1.00 2.00 2.00 3.24 5.00 6.00

## agency_lvl:

## numeric valued attribute

## attribute summary:

## Min. 1st Qu. Median Mean 3rd Qu. Max.

## 1.00 1.00 2.00 2.04 3.00 3.00

## lead_agency:

## numeric valued attribute

## attribute summary:

## Min. 1st Qu. Median Mean 3rd Qu. Max.

## 0.00 0.00 0.00 0.04 0.00 1.00

## tob_yrs:

## numeric valued attribute

## attribute summary:

## Min. 1st Qu. Median Mean 3rd Qu. Max.

## 1.00 3.00 4.50 6.76 9.00 21.00

## vertex.names:

## character valued attribute

## 25 valid vertex names

11.2 Building Exponential Random Graph Models 167

## No edge attributes

A quick examination of the network reveals that it is made up of three types of

organizations (local, state, and national), is made up of one connected component

that is fairly densely connected, and there is some variability of centrality across the

network members (shown both by the betweenness centralization score, as well as

the variability of the sizes of nodes in the ﬁgure, based on degree) (Fig. 11.1).

components(TCdiss)

## [1] 1

gden(TCdiss)

## [1] 0.343

centralization(TCdiss,betweenness,mode='graph')

## [1] 0.381

deg <- degree(TCdiss,gmode='graph')

lvl <- TCdiss %v% 'agency_lvl'

plot(TCdiss,usearrows=FALSE,displaylabels=TRUE,

vertex.cex=log(deg),

vertex.col=lvl+1,

label.pos=3,label.cex=.7,

edge.lwd=0.5,edge.col="grey75")

legend("bottomleft",legend=c("Local","State",

"National"),

col=2:4,pch=19,pt.cex=1.5)

11.2.1 Building a Null Model

It is often useful to start the modeling process by building a null model, one with no

substantive or structural predictors. This can be used as a baseline model to judge

how much subsequent models are improving. A null model typically only has one

term, edges, which produces a random graph model that has the same number of

edges as the observed network.

Fitting an ergm model uses syntax similar to other statistical modeling functions

in R. The ergm function is called with a model formula. This formula lists the

observed network as the dependent variable, and then all of the ergm model terms

are listed on the right hand side. Other options can also be set by the user. The

control option is used to pass control parameters to the ergm algorithm. Here we

168 11 Statistical Network Models

ITPC

Promotus

RTI

Quitline

IDHA

IN Latino Inst

King's Daughters

Hancock Regional

Hlthy Madison

Wabash

Tippecanoe

CHANCES

Johnson Memorial

ICSA

IN Cancer Consortium

Smokefree Pregnancies

ALA

AHA

DOE

DOH Diabetes Prev. & Chronic Disease

Medicaid

Partnership for Prev.

TTAC CTFK

ANR

Local

State

National

Fig. 11.1 Indiana tobacco control dissemination network

set the random number seed to ensure that the same results will be seen across

multiple runs. The results of ﬁtting the model are stored in a model object for further

examination and analysis.

library(ergm)

DSmod0 <- ergm(TCdiss ˜edges,

control=control.ergm(seed=40))

class(DSmod0)

## [1] "ergm"

summary(DSmod0)

## ==========================

## Summary of model fit

## ==========================

11.2 Building Exponential Random Graph Models 169

## Formula: TCdiss ˜ edges

## Iterations: 4 out of 20

## Monte Carlo MLE Results:

## Estimate Std. Error MCMC % p-value

## edges -0.648 0.122 0 <1e-04

## Null Deviance: 416 on 300 degrees of freedom

## Residual Deviance: 386 on 299 degrees of freedom

## AIC: 388 BIC: 392 (Smaller is better.)

A null model includes only the edges term. This acts as a type of intercept for

the model, and ensures that the simulated networks have the same number of edges

as the observed network. This can be seen by taking the logistic transformation

of the edges parameter, which gives the overall density of network. This demon-

strates that the null model is constrained by the number of edges in the observed

network.

plogis(coef(DSmod0))

## edges

## 0.343

11.2.2 Including Node Attributes

Once a null model is obtained, more interesting models can be ﬁt using a wide

variety of predictors. Following the order suggested by Table 11.1, we start with

main effect terms based on individual node characteristics. In the Indiana network,

we know which agency is the lead agency (receiving funding from CDC), and we

also know how long each agency has been working in tobacco control. It might

be reasonable to assume that agencies are more likely to be connected to the lead

agency. It also would make sense that agencies with a longer history of tobacco

control experience would be more likely to be connected to other agencies. The fol-

lowing scatterplot does suggest that there may be a relationship between experience

and interorganizational connections (Fig. 11.2).

scatter.smooth(TCdiss %v% 'tob_yrs',

degree(TCdiss,gmode='graph'),

xlab='Years of Tobacco Experience',

ylab='Degree')

170 11 Statistical Network Models

5 101520

5101520

Years of Tobacco Experience

Degree

Fig. 11.2 Basic association between years of experience and node degree

Both of the above hypotheses can be formally tested. There are two main

ergm model terms for testing node characteristic main effects, nodefactor and

nodecov.nodefactor is used for a categorical attribute (e.g., lead agency),

while nodecov is used for quantitative characteristics (such as tob yrs).

DSmod1 <- ergm(TCdiss ˜edges +

nodefactor('lead_agency')+

nodecov('tob_yrs'),

control=control.ergm(seed=40))

summary(DSmod1)

## ==========================

## Summary of model fit

## ==========================

## Formula: TCdiss ˜ edges +

## nodefactor("lead_agency") +

## nodecov("tob_yrs")

## Iterations: 16 out of 20

## Monte Carlo MLE Results:

## Estimate Std. Error

## edges -1.6783 0.3293

11.2 Building Exponential Random Graph Models 171

## nodefactor.lead_agency.1 17.9366 933.5551

## nodecov.tob_yrs 0.0599 0.0228

## MCMC % p-value

## edges 0 <1e-04

## nodefactor.lead_agency.1 0 0.9847

## nodecov.tob_yrs 0 0.0091

## Null Deviance: 416 on 300 degrees of freedom

## Residual Deviance: 323 on 297 degrees of freedom

## AIC: 329 BIC: 341 (Smaller is better.)

The results show a number of interesting points. First, the number of years in

tobacco control is positively and signiﬁcantly associated with the likelihood of obs-

erving a tie between the two agencies (remember that here a tie reﬂects a dissemi-

nation connection between two organizations). Second, although the parameter for

the effect of being the lead agency is quite large, it is not signiﬁcant. This appears

to be due to low precision in the estimate. With only one lead agency in a small

network of 25 members, there may not be the power to detect this effect. Finally,

the AIC (Akaike information criterion) for this model with two predictors is lower

than the AIC for the null model. This shows that this model is doing a better job of

explaining the data than the baseline model.

Similar to logistic regression analysis, we can estimate the probability of observ-

ing certain types of ties using the ﬁtted parameter estimates. This requires using the

logistic transformation to obtain numbers that can be properly interpreted as prob-

abilities. For example, based on Model 1, the following code calculates the prob-

ability that there is a dissemination tie between two agencies, one with 5 years of

tobacco control experience, the other with 10 years of experience, neither of whom

are the lead agency.

p_edg <- coef(DSmod1)[1]

p_yrs <- coef(DSmod1)[3]

plogis(p_edg +5*p_yrs +10*p_yrs)

## edges

## 0.314

The result is 0.31, which is just a little less than the overall density (i.e., overall

probability of observing a tie) of the dissemination network.

11.2.3 Including Dyadic Predictors

A rich source of hypotheses for network structures derive from questions about

homophily and heterophily. That is, are ties more or less likely between network

172 11 Statistical Network Models

members who are similar to each other on some characteristic (homophily) or

dissimilar (heterophily). This is a type of dyadic interaction predictor, and ergm

includes a number of these terms.

The raw frequencies of observed ties among and between different types of actors

in the network can be displayed with the mixingmatrix() function. Here, for

example, we see that the most frequent dissemination ties (24) are observed between

local and state-level agencies (see ?TCnetworks for the covariate code deﬁni-

tions). It can be somewhat complicated to interpret these raw frequency patterns;

however, they can generate hypotheses about dyadic interrelationships that can be

formally tested in the ERGM.

mixingmatrix(TCdiss,'agency_lvl')

## Note: Marginal totals can be misleading

## for undirected mixing matrices.

## 1 2 3

##1132414

##2241623

##3142313

mixingmatrix(TCdiss,'agency_cat')

## Note: Marginal totals can be misleading

## for undirected mixing matrices.

## 123456

## 1 0 12 3 2 3 4

##2121918 6 114

## 3 3 18 0 4 3 2

##4264116

##5313100

## 6 4 14 2 6 0 4

In the following models, the non-signiﬁcant lead agency predictor is dropped.

Three version of Model 2 are estimated, showing different options for including the

dyadic comparison of agency level as a predictor.

DSmod2a <- ergm(TCdiss ˜edges +

nodecov('tob_yrs')+

nodematch('agency_lvl'),

control=control.ergm(seed=40))

summary(DSmod2a)

## ==========================

## Summary of model fit

## ==========================

11.2 Building Exponential Random Graph Models 173

## Formula: TCdiss ˜ edges +

## nodecov("tob_yrs") +

## nodematch("agency_lvl")

## Iterations: 4 out of 20

## Monte Carlo MLE Results:

## Estimate Std. Error MCMC %

## edges -2.4808 0.3413 0

## nodecov.tob_yrs 0.1133 0.0201 0

## nodematch.agency_lvl 0.6875 0.2770 0

## p-value

## edges <1e-04

## nodecov.tob_yrs <1e-04

## nodematch.agency_lvl 0.014

## Null Deviance: 416 on 300 degrees of freedom

## Residual Deviance: 342 on 297 degrees of freedom

## AIC: 348 BIC: 359 (Smaller is better.)

Model 2a uses the basic nodematch term to include one network predictor

that assesses the effect on the likelihood of a dissemination tie when both organi-

zations are the same level (e.g., both are local organizations). This is a homophily

hypothesis, where we are testing if the same types of organizations are more likely

to communicate with one another. The positive and signiﬁcant parameter indicates

that there is a homophily effect here.

DSmod2b <- ergm(TCdiss ˜edges +

nodecov('tob_yrs')+

nodematch('agency_lvl',diff=TRUE),

control=control.ergm(seed=40))

summary(DSmod2b)

## ==========================

## Summary of model fit

## ==========================

## Formula: TCdiss ˜ edges +

## nodecov("tob_yrs") +

## nodematch("agency_lvl",

## diff = TRUE)

## Iterations: 4 out of 20

174 11 Statistical Network Models

## Monte Carlo MLE Results:

## Estimate Std. Error MCMC %

## edges -2.7792 0.3685 0

## nodecov.tob_yrs 0.1331 0.0217 0

## nodematch.agency_lvl.1 1.6145 0.4983 0

## nodematch.agency_lvl.2 -0.2148 0.3974 0

## nodematch.agency_lvl.3 1.3016 0.4422 0

## p-value

## edges <1e-04

## nodecov.tob_yrs <1e-04

## nodematch.agency_lvl.1 0.0013

## nodematch.agency_lvl.2 0.5891

## nodematch.agency_lvl.3 0.0035

## Null Deviance: 416 on 300 degrees of freedom

## Residual Deviance: 330 on 295 degrees of freedom

## AIC: 340 BIC: 358 (Smaller is better.)

Model 2b shows how to test a hypothesis of differential homophily. Here, in-

stead of one dyad term, there are three; one each for the three levels of the agency

level characteristic. Thus, this model now has three homophily terms, one for local

agencies, one for state, and one for national. The results suggest that the overall

homophily effect is seen mainly at the local and national levels.

DSmod2c <- ergm(TCdiss ˜edges +

nodecov('tob_yrs')+

nodemix('agency_lvl',base=1),

control=control.ergm(seed=40))

summary(DSmod2c)

## ==========================

## Summary of model fit

## ==========================

## Formula: TCdiss ˜ edges +

## nodecov("tob_yrs") +

## nodemix("agency_lvl", base = 1)

## Iterations: 4 out of 20

## Monte Carlo MLE Results:

## Estimate Std. Error MCMC %

## edges -1.1757 0.5372 0

11.2 Building Exponential Random Graph Models 175

## nodecov.tob_yrs 0.1340 0.0222 0

## mix.agency_lvl.1.2 -1.5805 0.5501 0

## mix.agency_lvl.2.2 -1.8354 0.6028 0

## mix.agency_lvl.1.3 -1.5363 0.5659 0

## mix.agency_lvl.2.3 -1.7073 0.5445 0

## mix.agency_lvl.3.3 -0.3110 0.6119 0

## p-value

## edges 0.0294

## nodecov.tob_yrs <1e-04

## mix.agency_lvl.1.2 0.0044

## mix.agency_lvl.2.2 0.0025

## mix.agency_lvl.1.3 0.0070

## mix.agency_lvl.2.3 0.0019

## mix.agency_lvl.3.3 0.6116

## Null Deviance: 416 on 300 degrees of freedom

## Residual Deviance: 330 on 293 degrees of freedom

## AIC: 344 BIC: 370 (Smaller is better.)

Finally, Model 2c shows how to include the most detailed tests for homophily and

heterophily. The nodemix term includes dyadic comparisons for all the possible

patterns of a categorical node attribute. The base option sets a reference category

for the effects, otherwise all possible effects are included (which may lead to model

stability problems). Here the reference category is (1,1), indicating the ties among

the local-level agencies. Note that for smaller networks and categorical attributes

with a large number of values, the mixing matrix may have empty cells (as well as

use up many more degrees of freedom). These can cause problems for the model

estimation and interpretation.

11.2.4 Including Relational Terms (Network Predictors)

The third type of predictor that can be included in ERGMs is a relational predictor,

where information about ties among the network members is used to predict the

likelihood of the dependent variable tie. This means that either other network vari-

ables can be used as predictors, or any other type of relational information among

the actors.

In this example, we use both a traditional network predictor and a relational

quantitative variable as predictors. First, we have the contact network variable,

which measured the frequency of contact among the Indiana tobacco control agen-

cies (1 = yearly; 2 = quarterly; 3 = monthly; 4 = weekly; and 5 = daily). Second, the

physical distance (in miles) between each pair of agencies was calculated and stored

in a statnet network object. For each of these predictors, the hypothesis is fairly

176 11 Statistical Network Models

evident. We expect that agencies who have more frequent contact with each other

will be more likely share information on the Best Practices guidelines. Second, we

also expect that the further apart two agencies are, the less likely they will be to

disseminate information to each other.

To include relational quantitative predictors in an ERGM, the edgecov term

can be used. Also, since these predictors are in the form of valued networks, the

attr option points to the edge attribute that contains the appropriate quantitative

information. First, we view a subset of the relational information for each of the

predictors, then the ERGM is ﬁt.

as.sociomatrix(TCdist,attrname ='distance')[1:5,1:5]

## 1 234 5

## 1 0.00 1.94 492 1870 1.27

## 2 1.94 0.00 492 1869 1.91

## 3 491.87 491.97 0 2325 493.08

## 4 1869.88 1868.98 2325 0 1868.61

## 5 1.27 1.91 493 1869 0.00

as.sociomatrix(TCcnt,attrname ='contact')[1:5,1:5]

## ITPC Promotus RTI Quitline IDHA

## ITPC 0 5 4 4 3

## Promotus 5 0 3 4 0

## RTI 4 3 0 2 0

## Quitline 4 4 2 0 0

## IDHA 3 0 0 0 0

DSmod3 <- ergm(TCdiss ˜edges +

nodecov('tob_yrs')+

nodematch('agency_lvl',diff=TRUE)+

edgecov(TCdist,attr='distance')+

edgecov(TCcnt,attr='contact'),

control=control.ergm(seed=40))

summary(DSmod3)

## ==========================

## Summary of model fit

## ==========================

## Formula: TCdiss ˜ edges +

## nodecov("tob_yrs") +

## nodematch("agency_lvl",diff = TRUE) +

## edgecov(TCdist, attr = "distance") +

## edgecov(TCcnt, attr = "contact")

11.2 Building Exponential Random Graph Models 177

## Iterations: 5 out of 20

## Monte Carlo MLE Results:

## Estimate Std. Error

## edges -4.850619 0.629666

## nodecov.tob_yrs 0.128750 0.028270

## nodematch.agency_lvl.1 1.795102 0.625698

## nodematch.agency_lvl.2 -0.646015 0.508164

## nodematch.agency_lvl.3 1.721850 0.546870

## edgecov.distance -0.000184 0.000253

## edgecov.contact 1.124253 0.146957

## MCMC % p-value

## edges 0 <1e-04

## nodecov.tob_yrs 0 <1e-04

## nodematch.agency_lvl.1 0 0.0044

## nodematch.agency_lvl.2 0 0.2046

## nodematch.agency_lvl.3 0 0.0018

## edgecov.distance 0 0.4683

## edgecov.contact 0 <1e-04

## Null Deviance: 416 on 300 degrees of freedom

## Residual Deviance: 237 on 293 degrees of freedom

## AIC: 251 BIC: 277 (Smaller is better.)

The results show that frequency of contact is positively associated with dissemi-

nation, as we hypothesized. On the other hand, physical distance between agencies

does not appear to be associated with dissemination.

11.2.5 Including Local Structural Predictors (Dyad Dependency)

The ﬁnal type of predictor that can be included in ERGMs is information about

local structural properties. Remember that we want to model how an entire network

looks and operates. By including some information about local structural tendencies

(such as the tendency for directed ties to be reciprocated), we can often produce

models that are much better ﬁts to the observed, empirical network. These types

of predictors lead to what are called dyadic-dependency models, and these present

many more computational and statistical challenges (Harris 2013). From the applied

analyst perspective, when using these types of predictors you should expect that the

execution time may go up dramatically, and that you are much more likely to run

into problems of model degeneracy and convergence failure.

In recent years, new types of local structural predictor terms have been discov-

ered and developed that at least partially avoid the most problematic model stability

178 11 Statistical Network Models

and convergence issues (Snijders and Pattison 2006). The three most commonly

used of these terms are GWDegree (geometrically weighted degree distribution),

GWESP (geometrically weighted edgewise shared partnerships), and GWDSP (geo-

metrically weighted dyadwise shared partnerships). See Harris’ monograph (2013)

for more details on how to choose and interpret these local structure predictors.

For our example, we include GWESP. This measures the effect of local clustering

(or transitivity) on the likelihood of observing a dissemination tie. The model results

do indicate that there is transitivity in the Indiana network, and that it is positively

associated with dissemination.

DSmod4 <- ergm(TCdiss ˜edges +

nodecov('tob_yrs')+

nodematch('agency_lvl',diff=TRUE)+

edgecov(TCdist,attr='distance')+

edgecov(TCcnt,attr="contact")+

gwesp(0.7,fixed=TRUE),

control=control.ergm(seed=40))

## Starting maximum likelihood estimation via MCMLE:

## Iteration 1 of at most 20:

## The log-likelihood improved by 0.5168

## Step length converged once.

## Increasing MCMC sample size.

## Iteration 2 of at most 20:

## The log-likelihood improved by 0.03238

## Step length converged twice. Stopping.

## This model was fit using MCMC.

## To examine model diagnostics and check

## for degeneracy, use the mcmc.diagnostics()

## function.

summary(DSmod4)

## ==========================

## Summary of model fit

## ==========================

## Formula: TCdiss ˜ edges + nodecov("tob_yrs") +

## nodematch("agency_lvl", diff = TRUE) +

## edgecov(TCdist, attr = "distance") +

## edgecov(TCcnt,attr = "contact") +

## gwesp(0.7, fixed = TRUE)

## Iterations: 2 out of 20

11.3 Examining Exponential Random Graph Models 179

## Monte Carlo MLE Results:

## Estimate Std. Error

## edges -6.326172 0.960886

## nodecov.tob_yrs 0.097673 0.029612

## nodematch.agency_lvl.1 1.557478 0.595676

## nodematch.agency_lvl.2 -0.266426 0.471595

## nodematch.agency_lvl.3 1.506054 0.526105

## edgecov.distance -0.000154 0.000243

## edgecov.contact 1.039252 0.145160

## gwesp.fixed.0.7 0.877169 0.377682

## MCMC % p-value

## edges 0 <1e-04

## nodecov.tob_yrs 0 0.0011

## nodematch.agency_lvl.1 0 0.0094

## nodematch.agency_lvl.2 0 0.5725

## nodematch.agency_lvl.3 0 0.0045

## edgecov.distance 0 0.5279

## edgecov.contact 0 <1e-04

## gwesp.fixed.0.7 0 0.0209

## Null Deviance: 416 on 300 degrees of freedom

## Residual Deviance: 231 on 292 degrees of freedom

## AIC: 247 BIC: 276 (Smaller is better.)

11.3 Examining Exponential Random Graph Models

11.3.1 Model Interpretation

The ﬁtted ERGM objects contain a lot of information about the parameter esti-

mates, simulated networks, and model ﬁt. The most important information is in-

cluded in the model summary output. Pay particular attention to any messages about

convergence issues, as they typically indicate estimation problems that should be

addressed.

The parameter estimates themselves, along with their standard errors, can be int-

erpreted similarly to logistic regression parameters. The p-values are associated with

the ratio of the parameter estimates to their standard errors, which are distributed

as Wald test statistics. Although not presented in the default summary output, the

individual parameter estimates can be exponentiated to produce odds-ratios.

The AIC and BIC values are related to overall model ﬁt, where lower numbers

indicate better ﬁt of the model to the observed network. Note how the AIC and BIC

180 11 Statistical Network Models

values were getting smaller as we progressively added predictors to the dissemina-

tion model. This is telling us that we were adding useful predictors to our ERGM.

Finally, as with any type of multivariate statistical model, it is often hard to und-

erstand the model by examining the individual parameter estimates. It is usually

helpful, then, to use the ﬁtted model to produce sets of forecasts that can be plot-

ted or examined to see how the model works across different proﬁles or ranges of

predictor values.

prd_prob1 <- plogis(-6.31 +2*1*.099 +1.52 +

4*1.042 +.858*(.50ˆ4))

prd_prob1

## [1] 0.408

prd_prob2 <- plogis(-6.31 +2*5*.099 +

1*1.042 +.858*(.50ˆ4))

prd_prob2

## [1] 0.0144

As a simple example, based on our ﬁnal model we predict the likelihood of a

dissemination tie being observed between two agencies, where they both have been

working in tobacco control for 1 year, they both are national-level agencies, they

have weekly contact, and the network has an average level of transitivity. We ignore

the effects of distance, given the small parameter value and lack of signiﬁcance. For

that predictor proﬁle, the probability of observing a dissemination tie is 41 %. In

comparison, for two agencies that have been working in tobacco control for 5 years,

that are at different levels (local and national, for example), and only have contact

yearly, the estimated probability is just 1.4 %. (See Harris 2013, for more in-depth

worked examples, as well as details on how the forecasting handles the local struc-

tural parameters such as GWESP here.)

11.3.2 Model Fit

The ergm package includes a number of tools that can be used to examine the ﬁt of

the network model to the data. First, make sure that there were no major problems

with convergence, and that the parameter values, standard errors, and p-values make

sense. AIC values can also be useful, especially examining how AIC (and BIC) is

reduced as more predictors are added to the model.

The simulations underlying the MCMC algorithms also provide useful infor-

mation for judging model ﬁt. The following procedure compares selected network

properties of the simulated networks based on our ﬁnal model to those same network

characteristics of the observed Indiana tobacco control network. Speciﬁcally, here

we will examine the geodesic distances, the distribution of edgewise shared part-

ners, the degree distribution, and the triad census (frequency of different patterns of

triangles).

11.3 Examining Exponential Random Graph Models 181

DSmod.fit <- gof(DSmod4,

GOF =˜distance +espartners +

degree +triadcensus,

burnin=1e+5,interval =1e+5)

summary(DSmod.fit)

## Goodness-of-fit for minimum geodesic distance

## obs min mean max MC p-value

## 1 103 88 102.90 116 1.00

## 2 197 123 169.44 200 0.04

## 3 0 0 8.90 34 0.20

## 4 0 0 0.02 1 1.00

## Inf 0 0 18.74 69 0.80

## Goodness-of-fit for edgewise shared partner

## obs min mean max MC p-value

## esp0 1 0 0.76 3 1.00

## esp1 8 0 4.94 14 0.28

## esp2 10 3 10.28 21 1.00

## esp3 7 7 17.68 34 0.02

## esp4 15 8 20.21 32 0.24

## esp5 11 7 16.77 25 0.32

## esp6 9 4 12.83 22 0.48

## esp7 14 3 7.73 16 0.12

## esp8 14 0 5.08 13 0.00

## esp9 5 0 2.77 8 0.32

## esp10 4 0 1.74 7 0.28

## esp11 1 0 1.19 3 1.00

## esp12 1 0 0.61 4 0.88

## esp13 2 0 0.13 1 0.00

## esp14 1 0 0.17 2 0.32

## esp15 0 0 0.01 1 1.00

## Goodness-of-fit for degree

## obs min mean max MC p-value

## 0 0 0 0.79 3 0.80

## 1 1 0 0.61 3 0.96

## 2 2 0 1.07 4 0.56

## 3 3 0 1.02 4 0.14

## 4 2 0 1.22 4 0.66

## 5 1 0 1.70 6 0.98

182 11 Statistical Network Models

## 6 2 0 1.87 5 1.00

## 7 2 0 2.47 7 1.00

## 8 0 0 2.76 7 0.14

## 9 1 0 2.57 8 0.54

## 10 3 0 2.25 6 0.84

## 11 2 0 2.15 5 1.00

## 12 1 0 1.45 5 1.00

## 13 1 0 1.05 4 1.00

## 14 2 0 0.62 3 0.20

## 15 1 0 0.22 2 0.40

## 16 0 0 0.13 1 1.00

## 17 0 0 0.05 1 1.00

## 18 0 0 0.08 1 1.00

## 19 0 0 0.11 1 1.00

## 20 0 0 0.24 1 1.00

## 21 0 0 0.22 1 1.00

## 22 0 0 0.24 1 1.00

## 23 0 0 0.09 1 1.00

## 24 1 0 0.02 1 0.04

## Goodness-of-fit for triad census

## obs min mean max MC p-value

## 0 832 616 756 911 0.20

## 1 759 787 881 944 0.00

## 2 517 407 502 625 0.72

## 3 192 114 161 221 0.18

The goodness-of-ﬁt object stores the results of those comparisons. If the model

is doing an adequate or good job of describing the observed network, then we would

expect to see that the simulated networks look like the observed network. For each

possible value of the selected network statistic, the frequency of the observed value

is reported alongside the minimum, mean, and maximum values across 100 (by

default) randomly simulated networks. The Monte Carlo empirical p-values are also

reported and these are the proportion of the simulated values of the statistic that

are at least as extreme as the observed value. Thus, small p-values (traditionally

<0.05) indicate cases where the model is not able to produce the particular network

characteristic (i.e., poor ﬁt).

Examination of DSmod.fit suggests that our relatively simple model with

seven predictors is doing a fairly good job of capturing the structural patterns in

TCdiss. Out of 50 network statistics, only four show poor ﬁt.

In addition to examining the actual goodness-of-ﬁt object, ergm can also easily

produce informative plots of the goodness-of-ﬁt. The resulting plot produces one

panel for each of the four network statistics. Each panel includes box-plots and

95 % empirical conﬁdence intervals (light grey lines) that show the variability of

11.3 Examining Exponential Random Graph Models 183

the individual network statistic across the simulated networks. The thick black line

indicates the value of the same statistic for the observed network. A good-ﬁtting

model will thus have the black line sitting inside the conﬁdence-range bands. In

Fig. 11.3 we can see that our ﬁnal model does a generally good job, but we can

also see that our model tends to produce networks that underestimate the number

of dyads with geodesics of two, while overestimating the dyads with geodesics of

three.

op <- par(mfrow=c(2,2))

plot(DSmod.fit,cex.axis=1.6,cex.label=1.6)

par(op)

11.3.3 Model Diagnostics

More detailed diagnostic information about the model estimation can be produced

with a call to the mcmc.diagnostics() function. This is useful for seeing

how the MCMC estimation process is running ‘under the hood,’ and is particularly

important if you run into convergence problems. The diagnostics report includes

statistical details for all of the covariates in the model, and also reports information

on how the model behaves over time. The plots produced display the MCMC chain

over time, and a resulting histogram. Both types of plots should show estimates that

are centered around 0 (Because of its length, only the diagnostics plots are shown

here.) (Fig. 11.4).

mcmc.diagnostics(DSmod4)

11.3.4 Simulating Networks Based on Fit Model

The ﬁtted ERGM can be used to produce one or more simulated networks that can

then be examined or analyzed as if it were an observed network. For example, this is

what a simulated network based on our ﬁnal model looks like compared to the obs-

erved network. Note that the simulated network will have the same node attributes

as the empirical network (Fig. 11.5).

sim4 <- simulate(DSmod4, nsim=1,seed=569)

summary(sim4,print.adj=FALSE)

## Network attributes:

## vertices = 25

## directed = FALSE

184 11 Statistical Network Models

123456 0 3 6 912 16

0.0 0.2 0.4 0.6

minimum geodesic distance

proportion of dyads

0.00 0.15 0.30

edge-wise shared partners

proportion of edges

0369 13 17

0.00 0.15 0.30

degree

proportion of nodes

0231

0.1 0.2 0.3 0.4

triad census

proportion of triads

Goodness-of-fit diagnostics

Fig. 11.3 Goodness-of-ﬁt plots for ﬁnal tobacco control model

## hyper = FALSE

## loops = FALSE

## multiple = FALSE

## bipartite = FALSE

## title = IN_Diffusion

## total edges = 115

## missing edges = 0

## non-missing edges = 115

## density = 0.383

## Vertex attributes:

## agency_cat:

## numeric valued attribute

## attribute summary:

## Min. 1st Qu. Median Mean 3rd Qu. Max.

## 1.00 2.00 2.00 3.24 5.00 6.00

11.3 Examining Exponential Random Graph Models 185

Sample statistics

−20 −10 0 10 20

0e+00 1e+06 2e+06 3e+06 4e+06

edges

0.00 0.02 0.04 0.06

−30 −20 −10 01020

edges

−50 5

0e+00 1e+06 2e+06 3e+06 4e+06

nodematch.agency_lvl.1

0.0 0.1 0.2 0.3

−550

nodematch.agency_lvl.1

−50 105

0e+00 1e+06 2e+06 3e+06 4e+06

nodematch.agency_lvl.2

0.00 0.05 0.10

−5−10 5010

nodematch.agency_lvl.2

−50 5

0e+00 1e+06 2e+06 3e+06 4e+06

nodematch.agency_lvl.3

0.00 0.05 0.10 0.15

−5−10 5010

nodematch.agency_lvl.3

−20000 0 10000

0e+00 1e+06 2e+06 3e+06 4e+06

edgecov.distance

0e+00 2e-05 4e-05 6e-05

−10000−20000 100000 20000

edgecov.distance

−400−200 0 200

0e+00 1e+06 2e+06 3e+06 4e+06

nodecov.tob_yrs

0.0000.0010.0020.0030.004

−400 −200 0 200 400

nodecov.tob_yrs

Fig. 11.4 MCMC diagnostics (partial)

186 11 Statistical Network Models

## agency_lvl:

## numeric valued attribute

## attribute summary:

## Min. 1st Qu. Median Mean 3rd Qu. Max.

## 1.00 1.00 2.00 2.04 3.00 3.00

## lead_agency:

## numeric valued attribute

## attribute summary:

## Min. 1st Qu. Median Mean 3rd Qu. Max.

## 0.00 0.00 0.00 0.04 0.00 1.00

## tob_yrs:

## numeric valued attribute

## attribute summary:

## Min. 1st Qu. Median Mean 3rd Qu. Max.

## 1.00 3.00 4.50 6.76 9.00 21.00

## vertex.names:

## character valued attribute

## 25 valid vertex names

## No edge attributes

op <- par(mfrow=c(1,2),mar=c(0,0,2,0))

lvlobs <- TCdiss %v% 'agency_lvl'

plot(TCdiss,usearrows=FALSE,

vertex.col=lvl+1,

edge.lwd=0.5,edge.col="grey75",

main="Observed TC network")

lvl4 <- sim4 %v% 'agency_lvl'

plot(sim4,usearrows=FALSE,

vertex.col=lvl4+1,

edge.lwd=0.5,edge.col="grey75",

main="Simulated network - Model 4")

par(op)

11.3 Examining Exponential Random Graph Models 187

Observed TC network Simulated network - Model 4

Fig. 11.5 Comparison of simulated network to observed tobacco control network

Chapter 12

Dynamic Network Models

What came ﬁrst–the music or the misery? Did I listen to the

music because I was miserable? Or was I miserable because I

listened to the music? Do all those records turn you into a

melancholy person? (Nick Hornby, High Fidelity.)

12.1 Introduction

Exponential random graph models, as presented in Chap. 11, allow for sophisticated

and powerful modeling of network structures and relationships. Generative models

of networks can be built using a wide variety of predictors, including node character-

istics, dyad characteristics, local structural characteristics, and even other network

relations. Substantive hypotheses can be tested with ERGM models, and estimated

models can be explored with the rich simulation and goodness-of-ﬁt tools that are

provided by the ergm package.

However, ERGMs are generally limited to cross-sectional network data. Social

networks, by their very nature, are dynamic. In particular, network ties are formed,

maintained, and sometimes dissolved over time. These dynamic ties processes may

be driven by a number of social processes, including characteristics of the actors,

dyads, and local network structures. For example, one student may become a friend

with another student partly because of the characteristics of the alter (e.g., attrac-

tiveness), partly because of their own similarity on some behavioral characteristic

(e.g., they both like the same type of music), or because of other local network struc-

tures (e.g., they both are already friends with the same other student). This chapter

covers stochastic actor-based models for network dynamics that are included in the

RSiena package, and which can be used to build models and test hypotheses about

networks as they change over time.

12.1.1 Dynamic Networks

Networks can change over time in two fundamental ways. First, networks can grow

or shrink over time, leading to changes in network composition. Second, as sug-

gested above, network ties can change among the network members. The modeling

methods discussed in this chapter apply primarily to changes of the second type.

D.A. Luke, A User’s Guide to Network Analysis in R, Use R!,

DOI 10.1007/978-3-319-23883-8 12

189

190 12 Dynamic Network Models

(Although RSiena can handle networks that have some changes in network compo-

sition, those changes themselves are not modeled.)

One of the challenges and opportunities for modeling network dynamics is that

overall network characteristics can often be the result of multiple underlying social

mechanisms. For example, homophily is the tendency for people (or other social

entities) to associate with others who are similar to them. Social networks tend to

exhibit strong homophily, and this pattern has been observed for decades across

many areas of social and health sciences. There are at least two social mechanisms

that can account for homophily – social selection and social inﬂuence. Social selec-

tion occurs when an actor selects or forms a new social tie with another actor who

is similar to her on some relevant characteristic. Social inﬂuence, on the other hand,

acts across existing social ties. Social inﬂuence occurs when the behavior of one

actor is changed to become more similar (or dissimilar) to the behavior of one or

more other actors.

Social

Selection

Social

Influence

Time 1 Time 2

Fig. 12.1 Comparison of social selection to social inﬂuence

Figure 12.1 visually depicts these two mechanisms. The colors designate some

node characteristic, such as smoking status. Blue nodes are smokers and green nodes

are non-smokers, for example. The top row illustrates social selection – the focal

actor (large blue node) starts out unconnected from the network. By time 2, the

actor has formed two new ties with other actors who are the same as him regard-

ing smoking status. The second row shows how social inﬂuence operates. Here, the

focal actor starts out connected to the network, but differs from his friends on smok-

ing status. At time 2 he has changed his behavior to match those of his friends. Note

that the time 2 networks are identical, and show strong homophily. However, we

see that this homophily can arise from two quite different mechanisms. Distinguish-

ing these mechanisms with real network data has important scientiﬁc and applied

consequences. For example, public health policies could use word-of-mouth com-

munication strategies to disseminate prevention messages across social networks.

12.1 Introduction 191

This policy would likely have greater effect if social inﬂuence mechanisms are the

primary dynamic, where social ties are already in place and messages are more

likely to pass from person to person. In contrast, if social selection is the primary

mechanism, then a word-of-mouth campaign may have less inﬂuence where indi-

viduals are choosing their new relationships based on behavioral similarity.

Disentangling the effects of social inﬂuence and social selection is not possible to

do with most network modeling techniques such as ERGM. This is mainly because

ERGMs are generally limited to cross-sectional network data. Accurately assessing

dynamic network mechanisms that operate over time requires dynamic modeling

techniques such as those in RSiena.

12.1.2 RSiena

SIENA stands for Simulation Investigation for Empirical Network Analysis, and is

a set of analytic tools that can be used to model longitudinal network data, accord-

ing to the stochastic actor-oriented model (SAOM) of Snijders and his colleagues

(Snijders et al. 2010). RSiena is the R package that contains the SIENA model

estimation functions, as well as a wide variety of supporting tools to plot, diagnose,

and examine the estimated models and simulated networks. The core SAOM is a

type of actor or agent-based simulation model – it uses estimation techniques fol-

lowing a Markov process that assumes that future changes in network states (i.e.,

formation or dissolution of a tie) are based probabilistically on the current state of

the complete network (Snijders et al. 2010).

RSiena combines the power of stochastic network modeling with longitudinal

analysis. This opens up many analytic possibilities. RSiena can be used to model

the evolution of one-mode networks, two-mode networks (see Chap. 9), and the

co-evolution of one-mode or two- mode networks with behavior. This last type of

model is what allows examination of social inﬂuence and social selection processes

in the joint evolution of friendship and smoking, for example.

With this power comes a fair amount of complexity. In particular, handling

missing data, analyzing longitudinal network data when there are changes in compo-

sition (i.e., nodes that enter or leave over time), selecting from hundreds of potential

parameter effects to include in the model, and dealing with estimation conver-

gence issues are all challenges that are beyond the ability of this short chapter to

handle in any detail. Instead, this chapter presents an introduction to RSiena anal-

ysis, using an example longitudinal network dataset that has been constructed to

help illustrate a basic approach to dynamic network modeling. This should help

readers get started using RSiena, but it is important to consult the many excel-

lent papers, tutorials, and documentation that are available. The manual for RSiena

is a good place to start, it is available at http://www.stats.ox.ac.uk/

˜snijders/siena/siena_r.htm.

192 12 Dynamic Network Models

12.2 Data Preparation

RSiena requires longitudinal (or panel) network data that have been collected from

the same network at two, or preferably more, timepoints. The Coevolve dataset

that is included in UserNetR has been developed to support exploration of a sim-

ple co-evolution model. The Coevolve data are in the form of a list of four igraph

networks. These are friendship networks among 37 students, measured at four points

in time (or waves). These data are based on an actual school-based directed friend-

ship network presented in Valente (2010). The original network was cross-sectional.

Here, we have added three additional ﬁctional waves of data that show changes both

in tie formation as well as changes in a ﬁctional smoking status variable.

In constructing the Coevolve networks, the following informal change rules were

used to create the new waves:

1. At each wave one smoker was randomly changed to non-smoker and three non-

smokers were changed to smokers, for a net gain of two smokers. The new

smokers were more likely to be network members who were connected to other

smokers.

2. At each wave, 10 % of the existing ties were randomly deleted. Then, the same

number of new ties were formed. This maintains the same overall density over

time.

3. When adding new directed ties, the following rules were used:

(a) Pick somebody who has the same smoking status

(b) Pick somebody who is popular (i.e., high indegree)

These informal rules were used to build in dynamics that are somewhat realistic,

but also simple enough to detect with a basic RSiena model. To examine the net-

works, they can be plotted using igraph. The data are stored as a list object, so

they should be extracted ﬁrst.

library(igraph)

library("UserNetR")

data(Coevolve)

fr_w1 <- Coevolve$fr_w1

fr_w2 <- Coevolve$fr_w2

fr_w3 <- Coevolve$fr_w3

fr_w4 <- Coevolve$fr_w4

Figure 12.2 shows the four waves of friendship data. Node shape conveys gen-

der (circle = female; square = male) and smoking status is conveyed by node color

(green = non-smoker; blue = smoker). The increase in smoking status over time is

fairly evident, and smoking status does seem to become more clustered, at least for

males.

12.2 Data Preparation 193

colors <- c("darkgreen","SkyBlue2")

shapes <- c("circle","square")

coord <- layout.kamada.kawai(fr_w1)

op <- par(mfrow=c(2,2),mar=c(1,1,2,1))

plot(fr_w1,vertex.color=colors[V(fr_w1)$smoke+1],

vertex.shape=shapes[V(fr_w1)$gender],

vertex.size=10,main="Wave 1",vertex.label=NA,

edge.arrow.size=0.5,layout=coord)

plot(fr_w2,vertex.color=colors[V(fr_w2)$smoke+1],

vertex.shape=shapes[V(fr_w2)$gender],

vertex.size=10,main="Wave 2",vertex.label=NA,

edge.arrow.size=0.5,layout=coord)

plot(fr_w3,vertex.color=colors[V(fr_w3)$smoke+1],

vertex.shape=shapes[V(fr_w3)$gender],

vertex.size=10,main="Wave 3",vertex.label=NA,

edge.arrow.size=0.5,layout=coord)

plot(fr_w4,vertex.color=colors[V(fr_w4)$smoke+1],

vertex.shape=shapes[V(fr_w4)$gender],

vertex.size=10,main="Wave 4",vertex.label=NA,

edge.arrow.size=0.5,layout=coord)

par(op)

Table 12.1 presents some basic descriptive statistics of the Coevolve network

data. As can be seen, the size and density of the networks remain constant, but the

number of smokers increases over time, as does the modularity based on smoking

status. Typically, detailed examination of network graphics and descriptive statistics

would happen prior to jumping into the dynamic modeling.

Wave Size Density Avg.InDegree Smokers Modularity

Wave 1 37 0.134 4.838 8 0.001

Wave 2 37 0.134 4.838 10 0.044

Wave 3 37 0.134 4.838 12 0.077

Wave 4 37 0.134 4.838 14 0.129

Table 12.1 Characteristics of Coevolve networks across four waves

The plots and descriptive statistics suggest that there are some network and be-

havioral dynamics that can be modeled using RSiena. In the rest of this chapter we

will explore a simple co-evolution model of friendship ties and smoking behavior.

The outline of the model building process has three main steps: (1) data preparation;

(2) model estimation; and (3) model exploration and testing.

Before starting, make sure to download and install the RSiena package. Like

most R packages, it is available through the CRAN repository. However, the RSiena

developers make newer versions of the package available at their website:

http://www.stats.ox.ac.uk/˜snijders/siena. This version, called

194 12 Dynamic Network Models

Wave 1 Wave 2

Wave 3 Wave 4

Fig. 12.2 Changes over time in Coevolve networks

RSienaTest, is the most up to date version and is used here. (For example, at

the time of writing this chapter RSiena from CRAN was version 1.1-232, while

RSienaTest was version 1.1-284.)

RSiena does not use the traditional method of specifying a statistical formula

for a modeling function, the way that ergm,lm, or in fact most other R statisti-

cal modeling procedures do. Instead, the modeling function (called siena07)is

applied to a set of RSiena objects, which must minimally include a data object

(containing all network and covariate data), an effects object (containing all of the

parameter effects to be included in the model), and an algorithm object (which

controls most of the modeling options).

The ﬁrst step, then, is packaging the network data in a way that RSiena can

understand. RSiena can handle six different types of variables. A network variable

is the basic dependent variable in an RSiena model, and can be a one-mode or

12.2 Data Preparation 195

two-mode network. A behavior variable is another type of dependent variable. It is

a node characteristic that changes over time, the evolution of which may considered

in a co-evolutionary model. In our example dataset, smoking status will be handled

as a behavior variable. Then there are four types of variables that are all handled as

covariates. A coCovar is a constant node attribute that does not change over time

(e.g., gender). A varCovar, on the other hand, is an attribute that does change over

time. (Note that a behavior variable is a type of varying covariate, but one that is

being treated as a dependent variable.) Similar to ergm,RSiena can also handle

dyadic covariates. A coDyadCovar is a constant dyadic covariate (for example, a

kinship relationship), while varDyadCovar is a dyadic covariate that changes over

time.

For our Coevolve data, we have one dependent network variable (the friendship

ties), one constant covariate (gender), and one varying covariate that will be handled

as a behavior dependent variable (smoking status). RSiena cannot handle igraph

or statnet data directly, it expects data in the form of raw arrays, matrices, or

vectors. So, some of the data management approaches covered in Chap. 3will be

useful here. The ﬁrst step is to transform the data into raw sociomatrices.

library(RSienaTest)

matw1 <- as.matrix(get.adjacency(fr_w1))

matw2 <- as.matrix(get.adjacency(fr_w2))

matw3 <- as.matrix(get.adjacency(fr_w3))

matw4 <- as.matrix(get.adjacency(fr_w4))

matw1[1:8,1:8]

## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]

## [1,] 0 0000011

## [2,] 0 0000010

## [3,] 0 1000000

## [4,] 0 0000010

## [5,] 0 0000000

## [6,] 0 0000010

## [7,] 1 0000101

## [8,] 1 0000010

Then, a dependent variable object is created with sienaDependent. This ex-

pects a stacked array with each array corresponding to one of the network waves. By

default, sienaDependent expects data in the form of a sparse matrix (from the

Matrix package). Here, the data are simple full matrices, so the sparse option

must be set to false.

fr4wav<-sienaDependent(array(c(matw1,matw2,matw3,matw4),

dim=c(37,37,4)),sparse=FALSE)

class(fr4wav)

## [1] "sienaDependent"

196 12 Dynamic Network Models

fr4wav

## Type oneMode

## Observations 4

## Nodeset Actors (37 elements)

The only problem with this approach is that these sociomatrices will get quite

large for larger networks. As discussed in Chap. 3, edgelists are preferable for han-

dling large networks. RSiena cannot natively handle edgelists, so they must be

transformed into sparse matrices as implemented by the Matrix package. The

following code produces the same friendship RSiena dependent variable, but via

edgelists instead of sociomatrices.

library(Matrix)

w1 <- cbind(get.edgelist(fr_w1), 1)

w2 <- cbind(get.edgelist(fr_w2), 1)

w3 <- cbind(get.edgelist(fr_w3), 1)

w4 <- cbind(get.edgelist(fr_w4), 1)

w1s <- spMatrix(37,37, w1[,1], w1[,2], w1[,3])

w2s <- spMatrix(37,37, w2[,1], w2[,2], w2[,3])

w3s <- spMatrix(37,37, w3[,1], w3[,2], w3[,3])

w4s <- spMatrix(37,37, w4[,1], w4[,2], w4[,3])

fr4wav2 <- sienaDependent(list(w1s,w2s,w3s,w4s))

fr4wav2

## Type oneMode

## Observations 4

## Nodeset Actors (37 elements)

Once the RSienda dependent variable is constructed, then other data objects

such as covariates can be created. Gender is stored in the igraph networks as

a vertex characteristic, so it is easy to extract that to create the coCoVar object.

Gender is coded 1 for female and 2 for males. The default for RSiena is to center

any covariate. This does not make much sense here, so it is turned off.

gender_vect <- V(fr_w1)$gender

table(gender_vect)

## gender_vect

## 1 2

## 22 15

gender <- coCovar(gender_vect,centered=FALSE)

gender

## [1]11112111121212221111121

## [24] 2 2 2 1 2 2 1 2 1 1 1 1 2 2

12.2 Data Preparation 197

## attr(,"class")

## [1] "coCovar"

## attr(,"centered")

## [1] FALSE

## attr(,"nodeSet")

## [1] "Actors"

Smoking status is our behavior variable for the co-evolution model. RSiena

expects an N×W matrix, with N (actor) rows and W (wave) columns. Again, we

extract this information from the igraph objects. Because a behavior variable is a

type of dependent variable, the sienaDependent function is used, but we specify

that this is a behavior variable.

smoke <- array(c(V(fr_w1)$smoke,V(fr_w2)$smoke,

V(fr_w3)$smoke,V(fr_w4)$smoke),dim=c(37,4))

smokebeh <- sienaDependent(smoke,type ="behavior")

smokebeh

## Type behavior

## Observations 4

## Nodeset Actors (37 elements)

Finally, all the individual variable objects are packaged together into a single

RSiena data object.

friend <- sienaDataCreate(fr4wav,smokebeh,gender)

friend

## Dependent variables: fr4wav, smokebeh

## Number of observations: 4

## Nodeset Actors

## Number of nodes 37

## Dependent variable fr4wav

## Type oneMode

## Observations 4

## Nodeset Actors

## Densities 0.13 0.13 0.13 0.13

## Dependent variable smokebeh

## Type behavior

## Observations 4

## Nodeset Actors

## Range 0 - 1

## Constant covariates: gender

198 12 Dynamic Network Models

Once this object is created then a basic descriptive report can be generated that

provides important information to examine prior to modeling.

print01Report(friend,modelname ='Coevolve Example' )

The results of this command are not directed to the console or saved in an R

object. Instead an external text ﬁle is created in the working directory, named in

this case ‘Coevolve Example.out’. It can be viewed using any text editor such as

Notepad. The report contains a variety of information about the RSiena data, inc-

luding information about any missing data, the degree distribution pattern observed

across the waves of network data, and summary information about each dependent

variable and covariate. A critical piece of information is found near the bottom of

the report under the heading ‘Change in Networks.’ The Jaccard index, which is a

measure of similarity, is calculated on the tie variables for each consecutive pair of

waves. Although there needs to be enough change between the observation periods

to allow for modeling, too much change would imply that the assumption of gradual

change is not tenable. The authors of RSiena suggest that Jaccard values should be

higher than 0.3 (Snijders et al. 2010). For the Coevolve data, we see Jaccard values

of greater than 0.8, suggesting a fairly high level of stability over time.

12.3 Model Speciﬁcation and Estimation

12.3.1 Speciﬁcation of Model Effects

Once the RSiena data have been put into the correct format, model speciﬁcation

and building can proceed. In Chap. 11 we saw that stochastic network models can

have a wide variety of parameters that test hypotheses about node attributes, simi-

larity of node attributes between dyads, tie attributes, and local network structural

properties. Longitudinal network models allow for an even larger set of potential

parameters, and choosing the theoretically appropriate set of parameters can be

challenging. In this chapter we will explore only a very small set of parameters.

For more detailed guidance, the RSiena documentation should be read closely

(especially Chap. 5– Model speciﬁcation).

The ﬁrst step for model speciﬁcation is to create an effects speciﬁcation object

with a minimal set of parameters.

frndeff <- getEffects( friend )

frndeff

## name effectName

## 1 fr4wav constant fr4wav rate (period 1)

## 2 fr4wav constant fr4wav rate (period 2)

## 3 fr4wav constant fr4wav rate (period 3)

12.3 Model Speciﬁcation and Estimation 199

## 4 fr4wav outdegree (density)

## 5 fr4wav reciprocity

## 6 smokebeh rate smokebeh (period 1)

## 7 smokebeh rate smokebeh (period 2)

## 8 smokebeh rate smokebeh (period 3)

## 9 smokebeh behavior smokebeh linear shape

## include fix test initialValue parm

## 1 TRUE FALSE FALSE 2.004 0

## 2 TRUE FALSE FALSE 2.004 0

## 3 TRUE FALSE FALSE 2.004 0

## 4 TRUE FALSE FALSE -0.807 0

## 5 TRUE FALSE FALSE 0.000 0

## 6 TRUE FALSE FALSE 0.208 0

## 7 TRUE FALSE FALSE 0.208 0

## 8 TRUE FALSE FALSE 0.208 0

## 9 TRUE FALSE FALSE 0.562 0

This is a basic model that only includes a small number of default effects, notably

the outdegree and reciprocity effects. Typically we will add other effects that we are

interested in testing or exploring. What are those effects and how are they speciﬁed?

All the effects that are available given the structure of the friend data set can be

seen using the effectsDocumentation function. Here is one place (of many)

where RSiena shows its non-R roots. Instead of sending the results to the R con-

sole, or creating a new R object, this function creates an HTML ﬁle that can then be

opened in your browser.

effectsDocumentation(frndeff)

The effects documentation report is a type of ‘effects dictionary’ that provides

all the information necessary for selecting appropriate model parameters. However,

the report can be daunting – for this simple example dataset with four timepoints,

one covariate (gender) and one behavior dependent variable (smokebeh), there are

over 400 possible effects!

Table 12.2 presents a set of effects that can be tested, along with the information

from the effects documentation that is used to correctly specify the effects in the

model estimation function. To understand this information, please also refer to the

‘frndeff.html’ ﬁle that is produced by the effectsDocumentation() function.

Because we are exploring a co-evolutionary model, we will examine effects on

the likelihood of tie formations (the fr4wav dependent variable), and the effects on

changes in behavior (the smokebeh dependent variable). So, there are two broad

types of effects, and they are distinguished by the ‘Name’ column in the effects doc-

umentation report. (‘ED Name’ in Table 12.2.) The actual speciﬁc effect that will be

included is speciﬁed by the term in the ‘shortName’ column. Many of these effects

will need to refer to a speciﬁc covariate or dependent variable, this is typically spec-

iﬁed by the term in the ‘interaction1’ column. To help navigate this information for

200 12 Dynamic Network Models

Effect Type ED name ED shortName ED interaction1 ED row

1 - Gender homophily Selection fr4wav sameX gender 139

2 - Ego smoking effect Selection fr4wav egoX smokebeh 200

3 - Alter smoking effect Selection fr4wav altX smokebeh 197

4 - Smoking homophily Selection fr4wav sameX smokebeh 212

5 - Avg. alter inﬂuence Inﬂuence smokebeh avSim fr4wav 321

6 - Total alter inﬂuence Inﬂuence smokebeh totSim fr4wav 324

7 - Reciprocity Structural fr4wav recip NA 14

8 - Transitivity Structural fr4wav transTrip NA 17

Table 12.2 RSiena effects for Coevolve data

the example, Table 12.2 also includes the particular row number from the complete

effects documentation report, but this will be accurate only if the data have been

prepared exactly the same way as in this chapter.

Eight different hypotheses will be tested with the eight effects listed in the table.

First, based on the pattern evident in Fig. 12.2, there appears to be a strong gender

homophily effect, where students are much more likely to be friends with other

students of the same gender. This is a type of social selection effect, we hypothesize

that the likelihood of an ego forming a new friendship tie is higher with an alter

who has the same gender. The RSiena term that will be used is ‘sameX.’ Next, two

different social selection main effects are hypothesized. The ﬁrst is the hypothesis

that based on the ego’s smoking status, he or she is more or less likely to form a

friendship tie (egoX). Conversely, the likelihood of forming a friendship tie may be

related to the smoking status of alters (altX). This may occur, for example, in schools

that have a strong pro-smoking culture. I may want to be a friend with somebody

because they smoke and smoking is ‘cool,’ regardless of whether I smoke or not.

Note that we have no reason to assume either of these hypotheses are true, given how

these data were constructed. Finally, the last social selection hypothesis is another

homophily effect, but this time in reference to smoking. The hypothesis is that new

friendship ties are more likely to be formed between two students who have the

same smoking status. Note that this hypothesis uses the same shortName.Inthis

case the interaction1 term is used to indicate that the homophily relationship

refers to smoking (instead of gender).

The next two hypotheses are about potential social inﬂuences on behavior. They

are similar to each other in that they focus on whether changes in smoking status can

be explained by patterns of smoking of the ego’s friends (to whom the ego is tied).

The ﬁrst hypothesis is that likelihood of changing behavior is related to the average

similarity of smoking status across all tied alters (avSim). The second hypothesis is

similar, but assumes that the inﬂuence is based on the total similarity across alters,

instead of average. (Total similarity captures the effect of having a larger personal

social network that inﬂuences behavior.)

The last two hypotheses are local structural effects. Here we will model the ten-

dency for friendship ties to be reciprocated (recip) as well as the general pattern of

transitivity (transTrip).

12.3 Model Speciﬁcation and Estimation 201

These effects are speciﬁed in an RSiena effects object. Typically, effects are

added one at a time using the includeEffects() function. The following code

adds the effects that correspond to the eight hypotheses just described.

frndeff <- getEffects( friend )

frndeff <- includeEffects(frndeff,sameX,

interaction1="gender",name="fr4wav")

## effectName include fix test initialValue

## 1 same gender TRUE FALSE FALSE 0

## parm

##10

frndeff <- includeEffects(frndeff,egoX,

interaction1="smokebeh",name="fr4wav")

## effectName include fix test initialValue

## 1 smokebeh ego TRUE FALSE FALSE 0

## parm

##10

frndeff <- includeEffects(frndeff,altX,

interaction1="smokebeh",name="fr4wav")

## effectName include fix test initialValue

## 1 smokebeh alter TRUE FALSE FALSE 0

## parm

##10

frndeff <- includeEffects(frndeff,sameX,

interaction1="smokebeh",name="fr4wav")

## effectName include fix test initialValue

## 1 same smokebeh TRUE FALSE FALSE 0

## parm

##10

frndeff <- includeEffects(frndeff,avSim,

interaction1="fr4wav",name="smokebeh")

## effectName include

## 1 behavior smokebeh average similarity TRUE

## fix test initialValue parm

## 1 FALSE FALSE 0 0

frndeff <- includeEffects(frndeff,totSim,

interaction1="fr4wav",name="smokebeh")

202 12 Dynamic Network Models

## effectName include

## 1 behavior smokebeh total similarity TRUE

## fix test initialValue parm

## 1 FALSE FALSE 0 0

frndeff <- includeEffects(frndeff,recip,transTrip,

name="fr4wav")

## effectName include fix test

## 1 reciprocity TRUE FALSE FALSE

## 2 transitive triplets TRUE FALSE FALSE

## initialValue parm

## 1 0 0

## 2 0 0

frndeff

## name effectName

## 1 fr4wav constant fr4wav rate (period 1)

## 2 fr4wav constant fr4wav rate (period 2)

## 3 fr4wav constant fr4wav rate (period 3)

## 4 fr4wav outdegree (density)

## 5 fr4wav reciprocity

## 6 fr4wav transitive triplets

## 7 fr4wav same gender

## 8 fr4wav smokebeh alter

## 9 fr4wav smokebeh ego

## 10 fr4wav same smokebeh

## 11 smokebeh rate smokebeh (period 1)

## 12 smokebeh rate smokebeh (period 2)

## 13 smokebeh rate smokebeh (period 3)

## 14 smokebeh behavior smokebeh linear shape

## 15 smokebeh behavior smokebeh average similarity

## 16 smokebeh behavior smokebeh total similarity

## include fix test initialValue parm

## 1 TRUE FALSE FALSE 2.004 0

## 2 TRUE FALSE FALSE 2.004 0

## 3 TRUE FALSE FALSE 2.004 0

## 4 TRUE FALSE FALSE -0.807 0

## 5 TRUE FALSE FALSE 0.000 0

## 6 TRUE FALSE FALSE 0.000 0

## 7 TRUE FALSE FALSE 0.000 0

## 8 TRUE FALSE FALSE 0.000 0

## 9 TRUE FALSE FALSE 0.000 0

## 10 TRUE FALSE FALSE 0.000 0

## 11 TRUE FALSE FALSE 0.208 0

12.4 Model Exploration 203

## 12 TRUE FALSE FALSE 0.208 0

## 13 TRUE FALSE FALSE 0.208 0

## 14 TRUE FALSE FALSE 0.562 0

## 15 TRUE FALSE FALSE 0.000 0

## 16 TRUE FALSE FALSE 0.000 0

12.3.2 Model Estimation

The last step before estimating the model is to set up any algorithm options that are

required. In this case, other than specifying a title string for the model, the algorithm

object will include all default options.

myalgorithm <- sienaAlgorithmCreate(projname='coevolve')

An RSiena model is estimated using the siena07() function, and by passing

the algorithm, data, and effects objects that were already created. Other options can

be speciﬁed as well to control the estimation process. In the following example, the

batch and verbose options are used to simplify the output. When run interac-

tively, you may want to set these options to TRUE. The last three options are used

to speed up the estimation process by using multiple cores of the computer’s CPU,

if available. Here, three cores, out of four, are used. Be careful about using all the

CPU cores, in case other processes are being run by the operating system. See the

help ﬁle for more technical details.

set.seed(999)

RSmod1 <- siena07( myalgorithm, data = friend,

effects = frndeff,batch=TRUE,

verbose=FALSE,useCluster=TRUE,

initC=TRUE,nbrNodes=3)

12.4 Model Exploration

12.4.1 Model Interpretation

As is typical with any R estimation technique, the results of the modeling can be

explored by viewing the contents of the model ﬁt object. Either list the RSiena ﬁt

object directly, or use the summary() command for a more detailed output. (The

formatting of the output listed here has been edited for length and legibility.)

204 12 Dynamic Network Models

summary(RSmod1)

## Estimates, standard errors and convergence t-ratios

## Network Dynamics

## 1. rate constant fr4wav rate (period 1)

## 2. rate constant fr4wav rate (period 2)

## 3. rate constant fr4wav rate (period 3)

## 4. eval outdegree (density)

## 5. eval reciprocity

## 6. eval transitive triplets

## 7. eval same gender

## 8. eval smokebeh alter

## 9. eval smokebeh ego

## 10. eval same smokebeh

## Estimate Standard Convergence

## Error t-ratio

## 1. 1.1572 ( 0.2073 ) -0.0491

## 2. 1.1410 ( 0.1990 ) -0.0114

## 3. 1.1366 ( 0.1948 ) -0.0273

## 4. -2.9556 ( 0.3949 ) 0.0141

## 5. 0.7990 ( 0.2539 ) -0.0805

## 6. 0.0860 ( 0.0785 ) -0.0508

## 7. 1.1429 ( 0.3174 ) -0.1147

## 8. 0.6885 ( 0.3792 ) -0.0122

## 9. -0.0847 ( 0.2878 ) 0.0165

## 10. 1.0975 ( 0.4373 ) -0.1392

## Behavior Dynamics

## 11. rate rate smokebeh (period 1)

## 12. rate rate smokebeh (period 2)

## 13. rate rate smokebeh (period 3)

## 14. eval behavior smokebeh linear shape

## 15. eval behavior smokebeh average similarity

## 16. eval behavior smokebeh total similarity

## Estimate Standard Convergence

## Error t-ratio

## 11. 0.3028 ( 0.1684 ) 0.0082

## 12. 0.3485 ( 0.1963 ) 0.0528

## 13. 0.3363 ( 0.1949 ) 0.0406

## 14. 4.0791 ( 8.7065 ) -0.0218

12.4 Model Exploration 205

## 15. 18.7278 ( 53.9223 ) -0.1372

## 16. -1.4614 ( 6.9254 ) 0.0824

## Total of 2340 iteration steps.

For a coevolution model, the parameter estimates are presented in two sections.

The network dynamics section contains the estimates pertaining to the tie formation

(i.e., the fr4wav dependent variable). Conversely, the behavior dynamics section

contains estimates related to changes in the network member behavior variable, here

it is smoking status.

The convergence t-ratios are not traditional t-statistics assessing the size of the

parameter estimates. Instead, they represent tests of the lack of convergence for each

estimate, so small values indicate good convergence. The RSiena manual suggests

that absolute values less than 0.10 indicate excellent convergence, and absolute val-

ues less than 0.15 are reasonable. Here we see that all of the network dynamics pa-

rameters have excellent convergence, while a few of the behavior parameters show

only reasonable convergence.

The rate estimates correspond to the estimated number of opportunities for

change per actor for each period (where period 1 is the time from wave 1 to wave

2). The eval estimates are the weights in the network evaluation function. The exact

calculations for a precise interpretation of the meaning of these effects is compli-

cated, see Snijders et al. (2010) for more details. However, they represent the relative

‘attractiveness’ of a particular network state for each actor. For example, the pos-

itive estimate for same gender (1.13) indicates that actors are more likely to form

new ties (or maintain existing ties) with other actors who have the same gender as

them.

The signiﬁcance of these evaluation function weights can be determined by di-

viding the estimates by their standard errors. These are distributed as t-statistics, so

any absolute values greater than 2 are signiﬁcant at the 0.05 signiﬁcance level.

For our example, we can see that our friendship formation is more likely with

alters who have the same gender and same smoking status as the ego. Conversely,

it appears that the main effects of ego smoking and alter smoking are not signiﬁ-

cant predictors of tie formation. Outdegree and reciprocity are signiﬁcant structural

predictors, but not transitivity. The behavior dynamics results suggest that smoking

status is increasing over time (linear shape). The large positive estimate for average

similarity would normally suggest that changes in smoking status are driven by the

overall similarity of the smoking status for all tied alters. However, this estimate has

a very large standard error, so the evaluation function weight is not being estimated

with much precision. This is not too surprising, because detecting changes in actor

behavior is harder than detecting changes in tie formation. There is greater power

to detect tie changes because of the greater number of potential ties (on the order

of the square of the size of the network for each period). Conversely, the number

of behavior changes is on the order of the simple number of network members for

each period. This is especially true for this example, where the network is relatively

206 12 Dynamic Network Models

small and only a few changes in smoking status happened at each wave. So, although

we know these smoking changes are ‘real’, the analysis does not have the required

power to detect the effects.

Typically, we will make adjustments and build subsequent models based on what

we learned from earlier models. For this example, we will drop a few non-signiﬁcant

predictors. To do this we simply update the effects object with either new predictors,

or by listing the predictors that we would like to drop. A dropped predictor is indi-

cated by the ‘include = FALSE’ option. Here we drop the total similarity predictor

for the behavior variable as well as the transitivity predictor.

frndeff2 <- includeEffects(frndeff,totSim,

interaction1="fr4wav",

name="smokebeh",

include=FALSE)

## [1] effectName include fix

## [4] test initialValue parm

## <0 rows> (or 0-length row.names)

frndeff2 <- includeEffects(frndeff2,transTrip,

name="fr4wav",

include=FALSE)

## [1] effectName include fix

## [4] test initialValue parm

## <0 rows> (or 0-length row.names)

frndeff2

## name effectName

## 1 fr4wav constant fr4wav rate (period 1)

## 2 fr4wav constant fr4wav rate (period 2)

## 3 fr4wav constant fr4wav rate (period 3)

## 4 fr4wav outdegree (density)

## 5 fr4wav reciprocity

## 6 fr4wav same gender

## 7 fr4wav smokebeh alter

## 8 fr4wav smokebeh ego

## 9 fr4wav same smokebeh

## 10 smokebeh rate smokebeh (period 1)

## 11 smokebeh rate smokebeh (period 2)

## 12 smokebeh rate smokebeh (period 3)

## 13 smokebeh behavior smokebeh linear shape

## 14 smokebeh behavior smokebeh average similarity

## include fix test initialValue parm

## 1 TRUE FALSE FALSE 2.004 0

## 2 TRUE FALSE FALSE 2.004 0

12.4 Model Exploration 207

## 3 TRUE FALSE FALSE 2.004 0

## 4 TRUE FALSE FALSE -0.807 0

## 5 TRUE FALSE FALSE 0.000 0

## 6 TRUE FALSE FALSE 0.000 0

## 7 TRUE FALSE FALSE 0.000 0

## 8 TRUE FALSE FALSE 0.000 0

## 9 TRUE FALSE FALSE 0.000 0

## 10 TRUE FALSE FALSE 0.208 0

## 11 TRUE FALSE FALSE 0.208 0

## 12 TRUE FALSE FALSE 0.208 0

## 13 TRUE FALSE FALSE 0.562 0

## 14 TRUE FALSE FALSE 0.000 0

myalgorithm <- sienaAlgorithmCreate(projname='coevol2')

Now the next model can be estimated. RSiena allows us to use the estimates

obtained from a previous model as the starting values for the new model estimation.

In this case we specify that the starting values should be based on the estimates

contained in RSmod1,usingtheprevAns option. This is also sometimes helpful

for improving the convergence of the individual weight estimates. Finally, in this

second model, we use the returnDeps option. This stores some required auxiliary

statistics on the simulated dependent variables for use in a subsequent exploration

of goodness-of-ﬁt.

set.seed(999)

RSmod2 <- siena07(myalgorithm,data = friend,

effects = frndeff2,

prevAns=RSmod1,batch=TRUE,

verbose=FALSE,useCluster=TRUE,

initC=TRUE,nbrNodes=3,

returnDeps=TRUE)

summary(RSmod2)

## Estimates, standard errors and convergence t-ratios

## Network Dynamics

## 1. rate constant fr4wav rate (period 1)

## 2. rate constant fr4wav rate (period 2)

## 3. rate constant fr4wav rate (period 3)

## 4. eval outdegree (density)

## 5. eval reciprocity

## 6. eval same gender

## 7. eval smokebeh alter

## 8. eval smokebeh ego

208 12 Dynamic Network Models

## 9. eval same smokebeh

## Estimate Standard Convergence

## Error t-ratio

## 1. 1.1392 ( 0.223 ) -0.0454

## 2. 1.1321 ( 0.213 ) 0.0056

## 3. 1.1260 ( 0.200 ) -0.0086

## 4. -3.0585 ( 0.422 ) 0.0695

## 5. 0.8387 ( 0.247 ) 0.0807

## 6. 1.4113 ( 0.303 ) 0.0423

## 7. 0.7123 ( 0.417 ) 0.0127

## 8. -0.0733 ( 0.307 ) 0.0721

## 9. 1.2041 ( 0.486 ) 0.0303

## Behavior Dynamics

## 10. rate rate smokebeh (period 1)

## 11. rate rate smokebeh (period 2)

## 12. rate rate smokebeh (period 3)

## 13. eval behavior smokebeh linear shape

## 14. eval behavior smokebeh average similarity

## Estimate Standard Convergence

## Error t-ratio

## 10. 0.3076 ( 0.170 ) 0.0508

## 11. 0.3680 ( 0.249 ) 0.0212

## 12. 0.3493 ( 0.181 ) 0.0178

## 13. 6.6261 ( 37.771 ) -0.0061

## 14. 18.6088 ( 111.033 ) 0.0270

## Total of 2140 iteration steps.

By dropping some non-signiﬁcant variables and starting with previously esti-

mated weight estimates, we have improved the convergence, now all effects have

excellent convergence. The similarity effect on smoking behavior is still positive,

but also still with a large standard error. A larger network, or more waves of data

would likely be required to improve the standard error.

12.4 Model Exploration 209

12.4.2 Goodness-of-Fit

Similar to the ergm package, RSiena includes graphical and diagnostic facilities

for exploring the goodness-of-ﬁt of an estimated model to the observed network. As

in ergm, goodness-of-ﬁt statistics can be calculated on properties of the simulated

networks that are not formally included as predictors in the ﬁtted models. This al-

lows us to assess the extent to which the model can produce simulated networks that

‘look like’ the observed network.

Goodness-of-ﬁt is assessed in RSiena using the sienaGOF function. A net-

work descriptive statistic is speciﬁed and is calculated across the simulated networks

at the end of each period. These values are then compared to the observed network

using the Mahalanobis distance.

In the current version of the RSienaTest package only a small number of

network statistics are built into the GOF function. One of these is the indegree dis-

tribution. In the following code, we also constrain the possible values of indegree to

the range 1–10 (using the levls option). This matches the range of indegrees in

the observed friendship network at wave 4.

table(degree(fr_w4,mode="in"))

##12345678910

##2376546211

gofi <- sienaGOF(RSmod2, IndegreeDistribution,

levls=1:10,verbose=FALSE,join=TRUE,

varName="fr4wav")

Once the GOF object is created, it can be used to plot the goodness-of-ﬁt infor-

mation. In Fig. 12.3, violin plots are used to display the variability of the descriptive

statistic across the simulated networks, in this case the indegrees. The dashed grey

lines represent the empirical 95 % conﬁdence interval. The red circles are the val-

ues of the descriptive statistic for the observed network. When the circles ﬁt inside

the conﬁdence intervals we interpret that as evidence of good ﬁt. In this case, our

second model does an excellent job of producing simulated networks that have the

same or similar indegree distributions.

plot(gofi)

Although only a few statistics are currently built into sienaGOF, RSiena sup-

ports adding in user-supplied statistic functions. This makes it easy to assess

goodness-of-ﬁt with almost any network characteristic that is of interest. The fol-

lowing example (taken from the sienaGOF-auxiliary help ﬁle) shows how to

deﬁne and then use the Holland and Leinhardt triad census (1978). The triad cen-

sus gives the frequency distribution of all possible triads in a directed network. The

census uses a 3-digit numeric code to designate one of the 16 possible patterns of

a triad. The ﬁrst digit indicates the number of reciprocated ties in the triad, the sec-

ond digit is the number of oneway ties, and the third digit is the number of empty

210 12 Dynamic Network Models

Goodness of Fit of IndegreeDistribution

105 106 109

123 45

p: 0.804

Statistic

789106

Fig. 12.3 Goodness-of-ﬁt for indegree

ties. So, 300 is the code for a triad connected by 3 reciprocal ties, while 003 is the

code for a completely unconnected triad. See Wasserman and Faust (1994)formore

information.

In the following code, a new TriadCensus function is created. This function

uses the existing triad.census function contained in the statnet package.

(Note that igraph could also be used to calculate descriptive statistics to be used

by sienaGOF.) This wrapper function is then used by sienaGOF.

TriadCensus <- function(i,data,sims,wave,

groupName,varName,levls=1:16){

unloadNamespace("igraph")

# to avoid package clashes

require(sna)

require(network)

x<- networkExtraction(i,data,sims,wave,

groupName,varName)

if (network.edgecount(x) <= 0){x<- symmetrize(x)}

# because else triad.census(x) will lead to an error

tc <- sna::triad.census(x)[1,levls]

# names are transferred automatically

12.4 Model Exploration 211

}

The GOF information stored in the ﬁt object can also directly examined or sum-

marized. The information can be plotted as before. For this type of auxiliary statistic

it is advisable in the plot to center and scale.

Here we see that our second model does a pretty good job of recreating the obs-

erved pattern of directed triads. It only fails in two cases: 021C (the model over-

estimates the number of this type of triad with two directed ties), and 120U (the

model underestimates this type of triad with one reciprocal tie and two directed ties)

(Fig. 12.4).

goftc <- sienaGOF(RSmod2, TriadCensus,

varName="fr4wav",

verbose=FALSE,join=TRUE)

descriptives.sienaGOF(goftc)

## 003 012 102 021D 021U 021C 111D

## max 12416 6506 4301 203 273 425 454

## perc.upper 12199 6248 4055 183 247 360 431

## mean 11830 5895 3775 152 208 302 390

## median 11832 5900 3774 151 208 301 390

## perc.lower 11474 5538 3513 124 173 252 350

## min 11289 5235 3369 111 153 199 328

## obs 11771 6018 3816 137 213 232 353

## 111U 030T 030C 201 120D 120U 120C

## max 336 100.0 19.00 182 80.0 54.0 85.0

## perc.upper 301 88.0 15.00 157 70.0 46.0 69.0

## mean 262 68.3 8.11 127 57.5 35.9 53.7

## median 263 68.0 8.00 126 58.0 36.0 54.0

## perc.lower 227 51.0 3.00 101 45.0 25.0 41.0

## min 214 39.0 1.00 91 41.0 19.0 35.0

## obs 245 82.0 8.00 119 61.0 49.0 65.0

## 210 300

## max 128 67.0

## perc.upper 118 55.0

## mean 102 43.3

## median 102 43.0

## perc.lower 85 32.0

## min 74 25.0

## obs 105 36.0

plot(goftc, center=TRUE,scale=TRUE)

212 12 Dynamic Network Models

Goodness of Fit of TriadCensus

p: 0.001

Statistic (centered and scaled)

11771

6018

3816

137

213

232 353

245

119

105

003 012 102 021D 021U 021C 111D 111U 030T 030C 201 120D 120U 120C 210 300

Fig. 12.4 Goodness-of-ﬁt for triad census

12.4.3 Model Simulations

The actual simulated networks can be accessed if the returnDeps option has

been set to true when estimating the model. (This can take a lot of memory for

larger networks, so the default is false.) The simulated network information is stored

in a nested list object (called sims) within the general ﬁtted model object. The

information is organized as an edgelist for each simulated network for each period.

The default number of simulated networks is 1,000 (set by the n3 parameter in

the sienaAlgorithmCreate function). Within a particular simulation run, one

predicted network is created at the end of each period. So, in this example, with

four waves of data there will be three simulated networks, one each for the ends of

periods one, two, and three.

The general structure of the sims information can be seen with the following

code. Here, the 500th simulation run is being accessed.

12.4 Model Exploration 213

str(RSmod2$sims[[500]])

## List of 1

## $ Data1:List of 2

## ..$ fr4wav :List of 3

## .. ..$ 1: int [1:188, 1:3] 1 1 2 2 2 2 3 3 3 3...

## .. ..$ 2: int [1:185, 1:3] 1 1 1 2 2 2 2 2 3 3...

## .. ..$ 3: int [1:176, 1:3] 1 1 1 2 2 2 2 2 2 2...

## ..$ smokebeh:List of 3

## .. ..$ 1: int [1:37] 1 0 0 0 0 1 0 0 0 0 ...

## .. ..$ 2: int [1:37] 1 1 0 0 0 1 0 0 0 0 ...

## .. ..$ 3: int [1:37] 1 0 0 1 0 1 0 0 0 0 ...

The actual edgelist can be obtained as follows, again for the 500th run. The

ﬁrst index speciﬁes the run number, the second index is the number of the group

(RSiena can do multiple-group estimation), the third index is the number of the

dependent variable, and the last index is the period number (or, equivalently, Wave -

1). For a coevolution model there are two dependent variables. The ﬁrst one is the tie

variable fr4wav, and the second is the behavior dependent variable smokebeh.

This code, then, provides the friendship tie edgelist for the 500th run and the third

period (limited to the ﬁrst 25 cases).

RSmod2$sims[[500]][[1]][[1]][[3]][1:25,]

## [,1] [,2] [,3]

## [1,] 1 7 1

## [2,] 1 8 1

## [3,] 1 11 1

## [4,] 2 7 1

## [5,] 2 17 1

## [6,] 2 21 1

## [7,] 2 30 1

## [8,] 2 32 1

## [9,] 2 35 1

## [10,] 2 37 1

## [11,] 3 13 1

## [12,] 3 17 1

## [13,] 3 20 1

## [14,] 3 21 1

## [15,] 3 33 1

## [16,] 4 13 1

## [17,] 4 17 1

## [18,] 4 21 1

## [19,] 4 27 1

## [20,] 5 10 1

## [21,] 5 22 1

214 12 Dynamic Network Models

## [22,] 5 25 1

## [23,] 5 28 1

## [24,] 5 29 1

## [25,] 6 9 1

The simulated smoking status information for the same run and wave is then:

RSmod2$sims[[500]][[1]][[2]][[3]]

## [1]10010100001001110000100

## [24] 1 0 0 1 0 0 0 0 1 1 1 0 0 1

Using the above information, you can access and transform the data into a

statnet or igraph network object. For example, the following code uses

igraph to create, examine, and plot one of the simulated networks from the second

RSiena model (Fig. 12.5).

library(igraph)

el <- RSmod2$sims[[500]][[1]][[1]][[3]]

sb <- RSmod2$sims[[500]][[1]][[2]][[3]]

fr_w4_sim <- graph.data.frame(el,directed =TRUE)

V(fr_w4_sim)$smoke <- sb

V(fr_w4_sim)$gender <- V(fr_w4)$gender

fr_w4_sim

## IGRAPH DN-- 37 176 --

## + attr: name (v/c), smoke (v/n), gender

## | (v/n), V3 (e/n)

## + edges (vertex names):

## [1] 1 ->7 1 ->8 1 ->11 2 ->7 2 ->17 2 ->21

## [7] 2 ->30 2 ->32 2 ->35 2 ->37 3 ->13 3 ->17

## [13] 3 ->20 3 ->21 3 ->33 4 ->13 4 ->17 4 ->21

## [19] 4 ->27 5 ->10 5 ->22 5 ->25 5 ->28 5 ->29

## [25] 6 ->9 6 ->11 6 ->15 6 ->34 7 ->1 7 ->8

## [31] 7 ->13 7 ->18 8 ->1 8 ->4 8 ->7 8 ->17

## [37] 8 ->19 8 ->21 8 ->30 8 ->32 9 ->6 9 ->7

## + ... omitted several edges

modularity(fr_w4_sim,membership =V(fr_w4_sim)$smoke+1)

## [1] 0.112

modularity(fr_w4,membership =V(fr_w4)$smoke+1)

## [1] 0.129

12.4 Model Exploration 215

colors <- c("darkgreen","SkyBlue2")

coord <- layout.kamada.kawai(fr_w4)

op <- par(mfrow=c(1,2),mar=c(1,1,2,1))

plot(fr_w4,vertex.color=colors[V(fr_w4)$smoke+1],

vertex.shape=shapes[V(fr_w4)$gender],

vertex.size=10,main="Observed - Wave 4",

vertex.label=NA,

edge.arrow.size=0.5,layout=coord)

plot(fr_w4_sim,

vertex.color=colors[V(fr_w4_sim)$smoke+1],

vertex.shape=shapes[V(fr_w4_sim)$gender],

vertex.size=10,main="Simulated - Wave 4",

vertex.label=NA,

edge.arrow.size=0.5,layout=coord)

par(op)

Observed - Wave 4 Simulated - Wave 4

Fig. 12.5 Comparison of observed network to simulated network

Chapter 13

Simulations

Sometimes, if you want to change a man’s mind, you have to

change the mind of the man next to him ﬁrst. (Megan Whalen

Turner – The King of Attolia)

13.1 Simulations of Network Dynamics

Chapter 10 illustrated how Rtools can be used to simulate networks with speciﬁc

structures, often based on particular network science models. These modeled net-

works are useful in that they reveal social structures that may reﬂect reality, or are

interesting for purely theoretical reasons. In any case, these are static networks.

However, social networks are dynamic. Social networks can grow or shrink over

time, and their composition can similarly change. For example, friendship networks

grow as people expand their friendship group, get smaller if friends move away or

friendships cool. The composition of friendship networks may change dramatically

during transitions (e.g., from middle school to high school or from college to work).

This type of network dynamics is captured by changes in node composition (which

people are in the network) and by changes in the pattern of ties connecting the nodes.

A second type of network dynamics is when some characteristic or behavior of

members in a social network is inﬂuenced by the structural properties of the network

itself. For example, in public health it has long been known that adolescents are

more likely to start smoking if they have friends or family members in their social

networks who smoke.

In the rest of this chapter two detailed examples will be presented that illustrate

how Rcan be used to model these two broad types of social dynamics. The abil-

ity to build simulations of network dynamics reﬂects a particular strength of R.In

particular, these network simulations are possible because of the integration of data

management, statistical programming, and network analysis in R. These types of

models are not possible to do in any traditional network analysis package such as

Pajek, UCINet, Gephi, or NodeXL.

D.A. Luke, A User’s Guide to Network Analysis in R, Use R!,

DOI 10.1007/978-3-319-23883-8 13

217

218 13 Simulations

13.1.1 Simulating Social Selection

When network structures change over time, we assume that there is an underlying

process of social selection that determines how new ties are formed or dissolved.

(Although this process may be at least partly random, we usually want to propose a

more interesting model where tie formation is driven by node characteristics, local

network structure, or network history.) Social selection has been studied extensively

in the social sciences, and it has been shown to partly explain homophily, which is

the general pattern that similar types of people tend to be connected to each other

(McPherson et al. 2001; also see Fig. 12.1).

In this section a dynamic model of social selection is constructed. The model is

deliberately kept simple, both to make it easier to understand, but also to reﬂect good

model building practices. In particular, it is generally a good idea when exploring

computational simulations to start simple, and only add complexity once the simple

model is fully understood. Also, this model is not built to emphasize efﬁciency

(speed of execution). Efﬁciency could be built in later when the simulation has been

fully tested and the analyst is ready to scale-up the simulation to handle larger runs.

For our model, we will assume that we have a friendship network where friend-

ship ties can change over time, and that these changes are driven by the similarity

(and dissimilarity) among network members on some abstract node characteristic.

This characteristic can be thought of as a behavior, or also possibly an attitude or

opinion. The characteristic is also quantitative, so that we can think of network

members having more or less of this characteristic. For example, the characteris-

tic might represent physical activity behavior, where some network members have

more of this characteristic (they exercise more) or less of it. In this example, we will

use the simulation model to understand how overall network homophily changes

over time based on individual changes in network tie formation and dissolution.

13.1.1.1 Setting Up the Simulation

To start building and testing our simulation, we need to have a network to work with.

We also want to deﬁne some basic network characteristics and model parameters

that will be used as we proceed. We will start with a simple random network that

has 25 members.

library(igraph)

N<- 25

netdum <- erdos.renyi.game(N, p=0.10)

graph.density(netdum)

mean(degree(netdum))

## [1] 0.1

## [1] 2.4

13.1 Simulations of Network Dynamics 219

The network also needs to include a node characteristic that will be used to drive

the network changes. This abstract behavior (Bh) can range from 0 to 1 to capture

diversity from low to high. To help with interpretation later on, a categorical node

characteristic (BhCat) is also calculated. The categorical variable has ﬁve levels,

and can be thought of as ‘very low,’ ‘somewhat low,’ ‘medium,’ ‘somewhat high,’

and ‘very high.’

Bh <- runif(N,0,1)

BhCat <- cut(Bh, breaks=5,labels =FALSE)

V(netdum)$Bh <- Bh

V(netdum)$BhCat <- BhCat

table(V(netdum)$BhCat)

##12345

##85246

Here is what the network looks like after setting up a color palette that maps onto

the ﬁve levels of BhCat. (See Chap. 5for details on how color palettes work.)

library(RColorBrewer)

my_pal <- brewer.pal(5,"PiYG")

V(netdum)$color <- my_pal[V(netdum)$BhCat]

crd_save <- layout.auto(netdum)

plot(netdum, layout = crd_save)

13.1.1.2 Creating an Update Function

To start the simulation building process, we will work from the inside out. That is,

we start by building the inner workings that allow a network to change, and then

build a simulation framework around that. The core of a dynamic network model is

allowing a network to change over time. Time is a rather abstract or slippery concept

in this context, but essentially we want to be able to change a network and observe

those changes. A good place to start is to deﬁne a function that updates the network

structure, in this case by forming or dissolving a single tie between two nodes.

Before the formal function is written, we can manually work through the process

by which we would like to form a new tie or dissolve an existing tie. Starting with

removing a tie is slightly easier. First, we need a way to identify all the existing ties

for a particular node. In igraph there are a couple of equivalent ways to extract the

adjacency list for a node in a network, which is a list of its direct ties. Here are the

nodes that are tied to Node 24 in netdum. (To save some syntax space and protect

us against inadvertent changes to the original data, the network is copied ﬁrst.)

220 13 Simulations

18 19

Fig. 13.1 Random test network with ﬁve levels of behavior indicated

g<- netdum

get.adjlist(g)[24]

## [[1]]

## + 4/25 vertices:

## [1] 2 9 15 22

g[[24,]]

## [[1]]

## + 4/25 vertices:

## [1] 2 9 15 22

Referring back to Fig. 13.1, you can see that Node 24 is connected to four other

nodes (2, 9, 15, and 22). The get.adjlist() syntax is easier to decipher than

the double-bracket shortcut, so that will be used hereafter.

In our model, we will want changes in ties to be driven by our node characteristic

(Bh). Speciﬁcally, a reasonable model might suppose that friendship ties are more

likely to be dissolved when the two friends are more dissimilar to each other on

the behavior of interest. To model this we want to be able to compare the behav-

ioral level of a particular network member with the behaviors of all of her friends.

This is easy to do, building on the previous syntax. This uses the igraph vertex

extractor function V(). Also, the get.adjlist() function returns a list, so the

unlist() function is used to obtain a simple numeric vector. Finally, the stored

adjacency list is used to ﬁlter the network to only display the behavior values for

the nodes adjacent to Node 24. It looks like Node 24 has low amounts of exercise,

which is similar to the level of Node 15. Node 24 is most dissimilar to Node 9, who

has a very high level of exercise.

13.1 Simulations of Network Dynamics 221

V(g)[24]$Bh

## [1] 0.192

V_adj <- unlist(get.adjlist(g)[24])

V(g)[V_adj]$Bh

## [1] 0.389 0.952 0.114 0.325

All of this can be combined into a simple dissimilarity vector that measures the

absolute values of the differences between the Bh values of Node 24 and its adjacent

nodes.

BhDiff <- abs(V(g)[V_adj]$Bh -V(g)[24]$Bh)

BhDiff

## [1] 0.1966 0.7604 0.0783 0.1327

The next step is to use this information to select a tie that should be removed from

the network. This can be done manually by identifying the two nodes and assigning

FALSE to the node pair (again, by making a copy ﬁrst):

gdum <- g

gdum[24,9]<- FALSE

get.adjlist(gdum)[24]

## [[1]]

## + 3/25 vertices:

## [1] 2 15 22

A more programmatic way to select the tie to remove is to identify the most

dissimilar pair of nodes. This can be done using index ﬁltering (Fig. 13.2).

gdum <- g

V_sel <- V_adj[BhDiff == max(BhDiff)]

gdum[24,V_sel] <- FALSE

get.adjlist(gdum)[24]

## [[1]]

## + 3/25 vertices:

## [1] 2 15 22

plot(gdum, layout = crd_save)

However, in our model we won’t always want to remove the tie from most dis-

similar pair of nodes. Instead, we would like to randomly remove ties where the

222 13 Simulations

18 19

Fig. 13.2 Test network with one tie (24-9) removed

probability is weighted by the amount of dissimilarity. This adds some randomness

and heterogeneity to the model. Once we have the dissimilarity vector, it can be used

to do this type of random tie selection.

gdum <- g

V_sel <- sample(V_adj,1,prob=BhDiff)

gdum[24,V_sel] <- FALSE

get.adjlist(gdum)[24]

## [[1]]

## + 3/25 vertices:

## [1] 2 15 22

The sampling function selects one node from the list of adjacent nodes, with

probability weighted by the dissimilarity vector. So, the more dissimilar the node

pair is, the more likely it will be selected to be removed. To check that the sampling

is working properly, we can sample multiple times and look at the selection distribu-

tion. This shows that Node 9 is selected most often, and Node 15 least often, which

matches expectations based on the similarity on Bh values.

smplCheck <- sample(V_adj,500,replace=TRUE,prob=BhDiff)

table(smplCheck)

## smplCheck

## 2 9 15 22

## 91 327 35 47

Adding a new tie from a particular node to any other node proceeds in a similar

fashion. The two main differences are that instead of picking the most dissimilar

13.1 Simulations of Network Dynamics 223

pair of nodes, we want to pick two nodes that are close to each other on the Bh

characteristic. Second, because we are wanting to add a new tie, we need a way

to select all of the non-adjacent nodes (i.e., nodes that are not directly tied to the

target node).

The non-adjacent nodes can be selected by removing the adjacent nodes as well

as the target vertex ID from a list of all the nodes. Nodes in igraph are numbered

from 1 to the total size of the network. So, if we are still interested in Node 24, the

following ﬁnds all non-adjacent nodes. This uses the vector indexing facility of R

where values are dropped if a negative sign is used.

vtx <- 24

nodes <- 1:vcount(g)

V_nonadj <- nodes[-c(vtx,V_adj)]

V_nonadj

## [1] 1 3 4 5 6 7 81011121314161718

## [16] 19 20 21 23 25

Following the same logic as before, we can now randomly create a new tie, based

on the similarity between a pair of nodes. The inverse of the absolute differences is

calculated, so now the vector BhDiff2 contains similarity scores.

BhDiff2 <- 1-abs(V(g)[V_nonadj]$Bh -V(g)[vtx]$Bh)

BhDiff2

## [1] 0.912 0.546 0.956 0.771 0.906 0.967 0.712

## [8] 0.900 0.207 0.663 0.859 0.309 0.997 0.194

## [15] 0.305 0.480 0.433 0.486 0.820 0.249

Sel_V <- sample(V_nonadj,1,prob=BhDiff2)

gnew <- g

gnew[vtx,Sel_V] <- TRUE

get.adjlist(gnew)[vtx]

## [[1]]

## + 5/25 vertices:

## [1] 2 9 10 15 22

The following code assigns a different color (“darkred”) to the newly added tie,

so that it can be seen easier in the plot (Fig. 13.3).

E(gnew)$color <- "grey"

E(gnew, P=c(vtx, Sel_V))$color <- "darkred"

plot(gnew, layout = crd_save)

All of this is preparation for creating a simple update function that can be called

within a larger network simulation. The function that follows accepts a network

(igraph) object and a target vertex. It ﬁrst checks to see if the passed vertex in the

224 13 Simulations

network is an isolate, if it is then the function silently returns the unaltered network

(because you can’t remove a tie from an isolate). If the vertex is not an isolate,

then the function randomly removes an existing tie with probability based on the

dissimilarity between all tied pairs. It then adds a new tie with probability based on

the similarity of all non-tied pairs. The two operations are combined into the same

function so that the returned network object has the same density (i.e., number of

total ties) as the original network. This makes the subsequent simulation easier to

interpret. Note that this function relies on the igraph object already having the

vertex attribute Bh deﬁned.

Sel_update <- function(g,vtx){

V_adj <- neighbors(g,vtx)

if(length(V_adj)==0)return(g)

BhDiff1 <- abs(V(g)[V_adj]$Bh -V(g)[vtx]$Bh)

Sel_V <- sample(V_adj,1,prob=BhDiff1)

g[vtx,Sel_V] <- FALSE

nodes <- 1:vcount(g)

V_nonadj <- nodes[-c(vtx,V_adj)]

BhDiff2 <- 1-abs(V(g)[V_nonadj]$Bh -V(g)[vtx]$Bh)

Sel_V <- sample(V_nonadj,1,prob=BhDiff2)

g[vtx,Sel_V] <- TRUE

}

18 19

Fig. 13.3 Test network with one new tie added to Node 24

To test the function, pass it an igraph object along with the vertex whose ties

are to be updated.

13.1 Simulations of Network Dynamics 225

gtst <- g

node <- 24

gnew <- Sel_update(g,node)

neighbors(gtst,node)

## + 4/25 vertices:

## [1] 2 9 15 22

neighbors(gnew,node)

## + 4/25 vertices:

## [1] 2 7 15 22

13.1.1.3 Building a Simple Simulation of Social Selection

Now that an update function is available that randomly adds and drops ties in the

abstract friendship network, a dynamic simulation model of social selection can be

built. The following simulation model is simple, but has all of the elements of a

dynamic network model. It operates on a network, makes changes over time, and

those changes are observable.

In a dynamic model, time can be realistic or highly abstract. For the social sel-

ection model presented here, an abstract notion of time is used. Speciﬁcally, nodes

will be selected randomly for updating, and we assume that there is some passage of

time between each update. However, no further speciﬁc characteristics of time are

provided or needed.

The following function Sel sim encapsulates the social selection simulation.

The function accepts an igraph network object and the number of desired updates.

It starts by deﬁning a list object that will be used to store the updated networks. Then

inside a loop that runs for the number of desired updates, a random node is selected

and the update function is called where an existing tie is removed, and a new tie

added. The updated network is stored in the list object after each step, and after the

loop is ﬁnished the entire network list is returned.

Sel_sim <- function(g,upd){

g_lst <- lapply(1:(upd+1), function(i)i)

g_lst[[1]] <- g

for (i in 1:upd) {

gnew <- g_lst[[i]]

node <- sample(1:vcount(g),1)

gupd <- Sel_update(gnew,node)

g_lst[[i+1]] <- gupd

}

g_lst

}

226 13 Simulations

This simulation function is inefﬁcient in a few ways. First, the loop could possi-

bly be replaced with a vectorized function. This could speed up the function, with

some loss in readability. More importantly, the function stores the entire network

for each update step. This will result in very large objects being returned by the

function, based on the size of the network and number of updates. It is helpful at

early stages of simulation development to preserve the whole networks, so that they

can be examined. However, in later stages it would be typical to change the func-

tion to only return speciﬁc information about the networks, rather than the networks

themselves.

The next step is to create a larger random network that will be used as input for

the social selection simulation.

N<- 100

netdum <- erdos.renyi.game(N, p=0.10)

graph.density(netdum)

## [1] 0.0949

mean(degree(netdum))

## [1] 9.4

Bh <- runif(N,0,1)

BhCat <- cut(Bh, breaks=5,labels =FALSE)

V(netdum)$Bh <- Bh

V(netdum)$BhCat <- BhCat

table(V(netdum)$BhCat)

##12345

## 13 19 32 13 23

Now that a starting network has been created, the actual simulation is run and the

results stored in a list of network objects. In this case, the simulation starts with the

netdum network object, and it is run for 500 updates. The returned list of network

objects contains the original network in the ﬁrst position, and then 500 additional

networks which correspond to each update in the simulation.

set.seed(999)

g_lst <- Sel_sim(netdum,500)

length(g_lst)

## [1] 501

summary(g_lst[[1]])

## IGRAPH U--- 100 470 -- Erdos renyi (gnp) graph

## + attr: name (g/c), type (g/c), loops (g/l),

## | p (g/n), Bh (v/n), BhCat (v/n)

13.1 Simulations of Network Dynamics 227

13.1.1.4 Interpreting the Results of the Simulation

The results of the simulation should always be examined ﬁrst to determine that the

simulation ran as expected, and then relevant characteristics of the networks can be

studied to see what patterns have emerged from the simulation modeling. Then it

can be determined if the results inform some research question or hypothesis about

the network dynamics.

A simple methods check for this simulation is that if the simulation worked as

intended, then we should see no changes in density over the simulated networks.

However, we should not see the same patterns of direct ties for any particular node

in the network. (Note that the tie patterns will only be different for a particular node

if that node was selected to be updated in the simulation. This is one reason to make

sure to run the simulation many more times than the size of the network, to help

ensure that all or at least most of the nodes have been updated.)

graph.density(g_lst[[1]])

## [1] 0.0949

graph.density(g_lst[[501]])

## [1] 0.0949

neighbors(g_lst[[1]],1)

## + 5/100 vertices:

## [1] 16 33 51 65 72

neighbors(g_lst[[501]],1)

## + 3/100 vertices:

## [1] 4 65 66

After it has been determined that the simulation is running properly, then more

substantive assessments can be done. A basic hypothesis for this simple example is

that we would expect the network to become more homophilous over time. This is

because the tie updates are partially driven by the similarities of the nodes on the

abstract behavioral characteristic. In this case, network modularity is a useful metric

(see Chap. 8for more information about modularity).

modularity(g_lst[[1]],BhCat)

## [1] -0.0221

modularity(g_lst[[501]],BhCat)

## [1] 0.168

Here we can see that the modularity was lower in the starting network compared

to the ﬁnal updated network. The higher modularity at the end tells us that ties

228 13 Simulations

between nodes in the same BhCat category are relatively more likely than ties

between different categories. That is, connected nodes are more similar to each other

than when the simulation started, thus demonstrating homophily.

This argument is more convincing if we use much more of the data provided by

the simulation. The following plots show that there is substantial variability of mod-

ularity across the steps of the simulation. More importantly, modularity increases

over time until near the end of the simulation run (Fig. 13.4).

sim_stat <- unlist(lapply(g_lst, function(u)

modularity(u,V(u)$BhCat)))

op <- par(mfrow=(c(1,2)))

plot(density(sim_stat),main="",xlab="Modularity")

plot(0:500,sim_stat,type="l",

xlab="Simulation Step",ylab="Modularity")

par(op)

13.1.2 Simulating Social Inﬂuence

This next example focuses on a second dynamic network process that can explain

homophily in social networks – namely, social inﬂuence. Social inﬂuence is the pro-

cess by which behaviors (or attitudes, opinions, etc.) of an individual are inﬂuenced

by the behaviors of those other persons who are close to them in their social net-

work. Here, we develop a simple model of this process where an abstract behavior

(Bh) for a particular member of a social network is inﬂuenced by the average beh-

aviors of all of those to whom the person is directly tied. This constitutes a simple

model of peer social inﬂuence. To make this slightly more realistic (and interesting)

we will also build into the model the concept of a tolerance region. Every member

of the network has a tolerance range (Tl). If an adjacent member has a value of

(Bh) that falls outside of the tolerance range, then that person’s behavior will not

inﬂuence them. So, for example, if we again think of Bh as a measure of the level

of physical activity, then in the following model the levels of physical activity of a

network member’s friends will inﬂuence her own level of activity, but only when

those friends levels of activity are somewhat close to her own.

13.1.2.1 Setting Up the Simulation

The model building process is similar to the previous example, so we can proceed

with slightly less exposition. The ﬁrst step is to set up an example network. The

only difference here is that we add a new vertex characteristic, Tl, the tolerance

range for each network member. To start with we assume that every member has the

same tolerance range of 0.20. (Remember that because of the random nature of the

simulation, your network values will not match what is presented below.)

13.1 Simulations of Network Dynamics 229

−0.05 0.05 0.15

01234567

Modularity

Density

0 100 200 300 400 500

0.00 0.05 0.10 0.15

Simulation Step

Modularity

Fig. 13.4 Modularity over time in the social selection simulation

N<- 25

netdum <- erdos.renyi.game(N, p=0.10)

Bh <- runif(N,0,1)

BhCat <- cut(Bh, breaks=5,labels =FALSE)

V(netdum)$Bh <- Bh

V(netdum)$BhCat <- BhCat

V(netdum)$Tl <- 0.20

V(netdum)$Tl[1:10]

## [1] 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

13.1.2.2 Creating an Update Function

The ﬁrst modeling step is once again to deﬁne a core update function. In this case,

instead of updating ties, for social inﬂuence the vertex attribute Bh needs to be

updated.

g<- netdum

V_adj <- neighbors(g,24)

V_adj

## + 4/25 vertices:

## [1] 2 9 15 22

230 13 Simulations

V(g)[24]$Bh

## [1] 0.192

V(g)[V_adj]$Bh

## [1] 0.389 0.952 0.114 0.325

For Node 24 in the example network netdum,Bh starts quite low (0.19). The

Bh values for Node 24’s four neighbors range from 0.11 to 0.95.

A simple way to update the Bh value for Node 24 would be to take the mean

of the starting value of Bh and the aggregate of the Bh values of all the neighbors.

The new Bh value is 0.32, but this has been adjusted by the Bh values for all four

neighbors. It has been particularly inﬂuenced by the high Bh value of Node 9 (0.95).

newval <- .5*(V(g)$Bh[24]+mean(V(g)[V_adj]$Bh))

newval

## [1] 0.318

The tolerance values can be used to ﬁlter out the nodes that have Bh values that

fall outside of the tolerance range. So, after setting up vectors that store the Bh

values, the neighbor Bh vector (NBh) is ﬁltered here to only show the values for

those neighbors whose Bh values are within 0.20 of Node 24’s Bh value.

V_Bh <- V(g)[24]$Bh

V_Bh

## [1] 0.192

N_Bh <- V(g)[V_adj]$Bh

N_Bh

## [1] 0.389 0.952 0.114 0.325

N_Bh[abs(N_Bh-V_Bh) <.20]

## [1] 0.389 0.114 0.325

So now an updated Bh value for Node 24 can be calculated after removing the

extremely high value for Node 9 that falls outside of the tolerance range. We see the

updated Bh value for Node 24 has not changed quite as much as before.

newval2 <- .5*(V(g)$Bh[24]+

mean(N_Bh[abs(N_Bh-V_Bh) <.20]))

newval2

## [1] 0.234

13.1 Simulations of Network Dynamics 231

This ﬁltering approach based on the absolute differences forms the heart of the

following social inﬂuence update function. This function returns an updated Bh

value for the selected vertex. If the selected vertex has no neighbors with Bh values

inside her tolerance region, the Bh value is returned unchanged. Once again, this

function assumes that the network object already has Bh and Tl vertex attributes.

Inf_update <- function(g,vtx){

TL <- V(g)[vtx]$Tl

V_adj <- neighbors(g,vtx)

V_Bh <- V(g)[vtx]$Bh

N_Bh <- V(g)[V_adj]$Bh

ifelse(length(N_Bh[abs(N_Bh-V_Bh)<TL]) >0,

new_Bh <- .5*(V_Bh +

mean(N_Bh[abs(N_Bh-V_Bh) <TL])),

new_Bh <- V_Bh

)

new_Bh

}

Testing out the new update function, it returns the correct value for Node 24.

newval3 <- Inf_update(g,24)

newval3

## [1] 0.234

13.1.2.3 Building the Simulation of Social Inﬂuence

Now that the social inﬂuence update function has been created, it can be put inside a

function that runs the dynamic network model. This is similar to the social selection

model in the previous section, with one big difference. It makes more sense to have

every node in the network be inﬂuenced at the same time, because we assume that

social inﬂuence is more of a continuous process. That means that instead of selecting

nodes randomly to get updated, here we will update every node in the network for

every run of the simulation. (Once again, the function presented here could be made

more efﬁcient in a number of ways.)

Inf_sim <- function(g,runs){

g_lst <- lapply(1:(runs+1), function(i)i)

g_lst[[1]] <- g

for (i in 1:runs) {

gnew <- g_lst[[i]]

232 13 Simulations

for (j in 1:length(V(g))) {

V(gnew)[j]$Bh <- Inf_update(g=gnew,vtx=j)

}

g_lst[[i+1]] <- gnew

}

g_lst

}

The following code sets up the same type of random starting network, and then

runs the inﬂuence model 50 times on the network. The simulation does not need to

be run as long as the selection model, because every node gets updated for every run

of the inﬂuence model.

N<- 100

netdum <- erdos.renyi.game(N, p=0.10)

Bh <- runif(N,0,1)

V(netdum)$Tl <- .20

V(netdum)$Bh <- Bh

V(netdum)$BhCat <- BhCat

set.seed(999)

g_lst <- Inf_sim(netdum,50)

13.1.2.4 Interpreting the Results of the Simulation

As the inﬂuence model runs, we might expect to see the variability of Bh to decrease

because of the way that network members are adjusting their behaviors based on the

average of their selected neighbors’ behaviors (Fig. 13.5).

op <- par(mfrow=(c(3,2)))

plot(density(V(g_lst[[1]])$Bh),xlim=c(-.2,1.2),

main="Original network")

plot(density(V(g_lst[[6]])$Bh),xlim=c(-.2,1.2),

main='After 5 runs')

plot(density(V(g_lst[[11]])$Bh),xlim=c(-.2,1.2),

main='After 10 runs')

plot(density(V(g_lst[[16]])$Bh),xlim=c(-.2,1.2),

main='After 15 runs')

plot(density(V(g_lst[[26]])$Bh),xlim=c(-.2,1.2),

main='After 25 runs')

plot(density(V(g_lst[[51]])$Bh),xlim=c(-.2,1.2),

main='After 50 runs')

par(op)

13.1 Simulations of Network Dynamics 233

−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0 0.4 0.8

Original network

N = 100 Bandwidth = 0.1054

Density

−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0 0.5 1.0 1.5

After 5 runs

N = 100 Bandwidth = 0.08352

Density

−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0 1.00.5 2.01.5

After 10 runs

N = 100 Bandwidth = 0.06472

Density

−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

02468

After 15 runs

N = 100 Bandwidth = 0.02197

Density

−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

0 50 150100

After 25 runs

N = 100 Bandwidth = 0.0009744

Density

−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

0 100000 250000

After 50 runs

N = 100 Bandwidth = 1.182e-06

Density

Fig. 13.5 Variability of Bh over time

The plots show us a few interesting things. First, it takes a few runs of the model

to start shifting the behavioral variability. However, the homogenization process

does not require 50 runs, by Run 25 almost all of the nodes have shifted their be-

havior to match those in the middle.

We can also see, however, that the network has become more homophilous. Fig-

ure 13.6 shows that by Run 25 most of the nodes fall into the middle Bh category,

whereas they started out evenly distributed among the ﬁve categories.

V(g_lst[[1]])$BhCat <- cut(V(g_lst[[1]])$Bh,

breaks=c(0,.2,.4,.6,.8,1), labels =FALSE)

V(g_lst[[26]])$BhCat <- cut(V(g_lst[[26]])$Bh,

breaks=c(0,.2,.4,.6,.8,1), labels =FALSE)

V(g_lst[[51]])$BhCat <- cut(V(g_lst[[51]])$Bh,

breaks=c(0,.2,.4,.6,.8,1), labels =FALSE)

234 13 Simulations

V(g_lst[[1]])$color <- my_pal[V(g_lst[[1]])$BhCat]

V(g_lst[[26]])$color <- my_pal[V(g_lst[[26]])$BhCat]

op <- par(mfrow=c(1,2),mar=c(0,0,2,0))

plot(g_lst[[1]],vertex.label=NA,

main="Original network")

plot(g_lst[[26]],vertex.label=NA,

main="Network after Run 25")

par(op)

These simulation examples are intended to provide simple examples to illustrate

the power of R for exploring network and behavioral dynamics. They can be ext-

ended in a number of ways, both to learn how to build such simulations, but also to

apply them to more serious scientiﬁc questions. There are at least two simple ways

to extend the examples presented here. First, in the social inﬂuence simulation in-

stead of having every network member start with the same tolerance range (0.20),

the effects of heterogeneous tolerance values on characteristics of social inﬂuence

could be explored. (Or, you can explore the effects of decreasing or increasing the

tolerance levels.) Second, the examples presented here separate out social inﬂuence

and social selection processes. The two simulations could be combined to explore

the characteristics of simultaneous selection and inﬂuence processes. (This is exam-

ined in Chap. 12 from a statistical modeling perspective.)

Original network Network after Run 25

Fig. 13.6 Greater homophily over time

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