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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
A comprehensive introduction to social network analysis that hones in on basic centrality measures, social links, subgroup analysis, data sources, and more Written by military, industry, and business professionals, this book introduces readers to social network analysis, the new and emerging topic that has recently become of significant use for industry, management, law enforcement, and military practitioners for identifying both vulnerabilities and opportunities in collaborative networked organizations. Focusing on models and methods for the analysis of organizational risk, Social Network Analysis with Applications provides easily accessible, yet comprehensive coverage of network basics, centrality measures, social link theory, subgroup analysis, relational algebra, data sources, and more. Examples of mathematical calculations and formulas for social network measures are also included. Along with practice problems and exercises, this easily accessible book covers: * The basic concepts of networks, nodes, links, adjacency matrices, and graphs * Mathematical calculations and exercises for centrality, the basic measures of degree, betweenness, closeness, and eigenvector centralities * Graph-level measures, with a special focus on both the visual and numerical analysis of networks * Matrix algebra, outlining basic concepts such as matrix addition, subtraction, multiplication, and transpose and inverse calculations in linear algebra that are useful for developing networks from relational data * Meta-networks and relational algebra, social links, diffusion through networks, subgroup analysis, and more An excellent resource for practitioners in industry, management, law enforcement, and military intelligence who wish to learn and apply social network analysis to their respective fields, Social Network Analysis with Applications is also an ideal text for upper-level undergraduate and graduate level courses and workshops on the subject.
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Seitenzahl: 389
Veröffentlichungsjahr: 2013
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Table of Contents
Title Page
Copyright
series
List of Figures
List of Tables
Foreword
Preface
Acknowledgments
Introduction
Chapter 1: What is a Network?
1.1 Basic Network Concepts
1.2 Adjacency Matrices, Graphs, and Notation
1.3 Nodes and Links
1.4 Good will Hunting Problem
1.5 Formal and Informal Networks
1.6 Summary
Chapter 1 Lab Exercise
References
Chapter 2: Centrality Measures
2.1 What is “Centrality” and Why do we Study IT?
2.2 Calculating Nodal Centrality Measures
2.3 Directed Networks and Centrality Measures
2.4 Location in the Network
2.5 Summary
Chapter 2 Lab Exercise
References
Chapter 3: Graph Level Measures
3.1 Density
3.2 Diameter
3.3 Centralization
3.4 Average Centralities
3.5 Network Topology
3.6 Summary
Chapter 3 Lab Exercise
References
Chapter 4: Social Links
4.1 Individual Actors
4.2 Social Exchange Theory
4.3 Social Forces
4.4 Graph Structure
4.5 Agent Optimization Strategies in Networks
4.6 Hierarchy of Social Link Motivation
4.7 Summary
References
Chapter 5: Subgroup Analysis
5.1 Subgroups
5.2 Organizational Theory
5.3 Random Groups
5.4 Heuristics for Subgroup Identification
5.5 Analysis Methods
5.6 Summary
Chapter 5 Lab Exercise
References
Chapter 6: Diffusion and Influence
6.1 Applications for Social Diffusion
6.2 Strain Theory
6.3 Social Context
6.4 Group Impacts on Diffusion
6.5 Network Structure and Diffusion
6.6 Group Influence Strategies and Bases of Power
6.7 Summary
References
Chapter 7: Meta-Networks and Relational Algebra
7.1 Modes of Data
7.2 Source, Target, Direction
7.3 Multimode Networks
7.4 Bridging a Meta-Network
7.5 Strength of Ties
7.6 Summary
Chapter 7 Lab Exercise
References
Chapter 8: Sources of Data
8.1 Network Sampling
8.2 Measuring Links
8.3 Data Quality
8.4 Additional Ethnographic Data Collection Methods
8.5 Anonymity Issues
8.6 Summary
References
Chapter 9: Organizational Risk
9.1 What is Risk?
9.2 Measures of Centrality and Risk
9.3 Other Risk Measures
9.4 The Right Network: Efficient Versus Learning/Adaptive
9.5 Network Threats and Vulnerabilities
9.6 Thickening a Network
9.7 Thinning a Network
9.8 Process of Organizational Risk Analysis
9.9 Summary of Main Points
Chapter 9 Lab Exercise
References
Appendix A: Matrix Algebra Primer
Appendix B: Tables of Data and Networks
Appendix C: Five Points of a Graph
Index
Copyright © 2013 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data:
Social Network Analysis with Applications / McCulloh . . . [et al.].
``Wiley-Interscience.''
Includes bibliographical references and index.
ISBN 978-1-118-16947-6 (pbk.)
1. Networks- Analysis. 2. Social
sciences- Research- Statistical methods. I. McCulloh II. Armstrong III. Johnson
HA31.2.S873 2007
001.4'33- dc22
2004044064
To Our Students
List of Figures
I.1
Emotions mapped by new geography
xxvii
1.1
Graph G
3
1.2
A social network of 10 people
5
1.3
(a) Undirected and (b) directed graphs
7
1.4
Good Will Hunting solution
11
1.5
Edge list and network diagram for homonyms
12
1.6
Edge list and network diagram for four letters in common
12
1.7
Edge list and network diagram for three letters in common
13
1.8
Edge list and network diagram for two letters in common
13
1.9
R&D organization formal structure
14
1.10
R&D Organization informal structure
14
1.11
Formal NCO chain of support
15
1.12
Informal NCO network
16
2.1
Core periphery graph
30
2.2
Not-so-neat network
30
2.3
Comparing graphs with nodes sized by (a) degree centrality and (b) eigenvector centrality
31
2.4
Small world network with nodes sized by betweenness centrality
32
2.5
A scale-free network with nodes sized by degree centrality, and isolate nodes are hidden
32
2.6
(a) Star and (b) circle graphs
34
2.7
Euler bridge problem: (a) map of Konigsberg and (b) graph of Konigsberg bridges
35
2.8
Example graph for betweenness calculation
36
2.9
Weighted graph
41
2.10
Network of eight hyperlinked web pages
44
2.11
Directed graph
46
2.12
Network graph for the research organization
47
2.13
Network graph for the research organization
50
2.14
Network graph with the link between John and Fred removed
51
2.15
Centrality example graph
67
3.1
Network density chart
70
3.2
Star graph
72
3.3
Circle graph
72
3.4
Two different networks: (a) network A and (b) network B
73
3.5
The lattice network topology
80
3.6
The mesh pattern in a lattice network
80
3.7
Small world network topology
81
3.8
Core-periphery network of 100 nodes
83
3.9
Cellular network of 100 nodes
84
3.10
Scale-free network of 100 nodes
84
3.11
Scale-free network of 100 nodes
85
4.1
Power in basic network exchanges: (a) two-node network, (b) three-node network, and (c) four-node network
112
4.2
Eight possible signed triads
118
4.3
Two signed graphs: (a) unbalanced network and (b) balanced network
120
4.4
Pareto optimality space
123
4.5
Four stages of link formation
124
5.1
Example network
135
5.2
Hierarchical clustering chart produced in ORA for the example network of Figure 5.1
138
5.3
Block model chart produced in ORA for the example network of Figure 5.1
138
5.4
Block reduced chart produced in ORA for the example network of Figure 5.1
139
5.5
Example network for Newman grouping
140
5.6
Network with edge labels
140
5.7
Highest betweenness edge removed
141
5.8
Newman subgroup emerges
141
5.9
Iterate of Newman algorithm
141
5.10
Iterate of Newman algorithm
142
5.11
Newman groups
142
5.12
Empty hierarchical clustering diagram
143
5.13
Hierarchical clustering diagram begins to fill
143
5.14
Full hierarchical clustering diagram
143
6.1
Moore's chasm and Rogers' diffusion curve [after Moore, 1991; Rogers et al., 2005]
158
6.2
Actors involved in the diffusion process
159
6.3
Opinion leaders in a network
160
7.1
Multimode network of agents and resources
174
7.2
Source to target with matrix
X
175
7.3
Source to target with matrix
X
and its transpose
175
7.4
Source to target calculation for bidirectional link
176
7.5
Two node types: agents and resources
177
7.6
Dot product of networks X and Y: agent to resource to agent
178
7.7
Dot product of networks X and Y: both agent to resource
179
7.8
All possible combinations linking agents to a resource
180
7.9
Bridging the gap between resources in network Z
181
9.1
Undirected network graph
208
9.2
Social network before the removal of Bill
211
9.3
Fragmented social network after the removal of Bill
211
9.4
Removal of the link between Bill and Alan
212
9.5
John and Lee make a direct link
213
9.6
Linking of knowledge and resources for Task 1
213
9.7
The network showing the spread of an interesting image by e-mail
218
9.8
The three most critical nodes to the diffusion across the network, nodes 3, 8, and 65
218
9.9
The three most critical nodes if removed would most fragment the network, nodes 8, 52, and 69
219
9.10
Cellular structure associated with clandestine networks
219
9.11
The efficient structure of InterSec Pty Ltd. in the agent × agent network
221
9.12
Total network for InterSec Pty Ltd. as an efficient organization structure
222
9.13
Learning organization structure in the Agent × Agent network for InterSec Pty Ltd.
223
9.14
New total network for InterSec after changing structure to a learning organization
224
9.15
Activities in identifying risks in organizations
229
List of Tables
1.1
Adjacency Matrix of a Friendship Graph
5
1.2
Some Common Notations Used to Represent Networks Mathematically
6
1.3
UnDirected Matrix
7
1.4
Directed Matrix
7
1.5
3-Step Walks from Node 1 to Node 2
10
2.1
Star Graph Matrix
35
2.2
List of Geodesic Paths
37
2.3
Geodesic Path Calculation for Numerator of Agent_2
38
2.4
Adjacency Matrix for the Research Organization
48
2.5
Summary of the Centrality Measures
48
2.6
Comparisons of the Centrality Measures
50
2.7
Comparisons of the Centrality Measures
52
3.1
Comparison of Network Topologies
79
4.1
Stages of Social Link Motivation and Stages of Small-Group Development
125
5.1
Adjacency Matrix for Example Graph
136
5.2
Newman Grouping
136
5.3
CONCOR Grouping
136
5.4
External/Internal Link Analysis
140
8.1
Caleb Methodology Table
198
9.1
Unscaled Centrality Measures
209
9.2
Geodesic Path Calculation for Ellen, Bill, and Alan
214
B.1
Lab Exercise 1-Agent Node Class
241
B.2
Lab Exercise 1-Agent×Agent Adjacency Matrix
241
B.3
Health Organization-Agent Node Class
242
B.4
Health Organization-Agent×Agent Admin Advice
243
B.5
Health Organization-Agent×Agent Clinical Advice
249
B.6
Dolphin Agent Names
255
B.7
Dolphins Agent × Agent Network
257
B.8
Lab Exercise 5-Subgroup1 Agent×Agent Network
263
B.9
Lab Exercise 5 Agent Attributes
268
B.10
Lab Exercise 7-Agent×Location Network
268
B.11
Lab Exercise 7-Agent×Resource Network
269
B.12
Lab Exercise 7-Resource×Agent Network
269
B.13
Lab Exercise 7-Location×Location Network
269
B.14
Lab Exercise 7-Resource×Resource Network
270
B.15
Lab Exercise 9-Formal Social Network: Agent×Agent
270
B.16
Lab Exercise 9-Informal Social Network: Agent×Agent
271
B.17
Lab Exercise 9-Agent×Knowledge Network Indicating Which Agent Holds What Knowledge.
271
B.18
Lab Exercise 9-Agent×Task Network Indicating the Task Each Agent has been Assigned.
272
B.19
Lab Exercise 9-Knowledge×Task Network Showing What Knowledge is Required to Carry Out Which Tasks.
272
Foreword
Readers of this book probably already realize that both popular attention to social networks and the systematic study of networks have exploded in the past decade or so. What might be less apparent is the historical depth and substantive breadth of the network perspective. Tracking down the very first scholarly use of the word “network” is much like locating the first living creature on earth—it had to be there somewhere, but once life (or the study of networks, for that matter) emerged, expansion and diversity took over and, in turn, provided the impetus for increasing diversity. The study of networks has followed a similar trajectory of diversification. As a maturing arena of inquiry, the field of networks has expanded from a somewhat arcane branch of mathematics (graph theory) and a relatively focused structural approach in the social sciences (sociometry and its descendants) to a powerful perspective for studying relational systems quite generally. Applications currently extend into essentially all domains of investigation: social structure and process, biological systems, formal organizations, computer networks, physical connections, semantic structures, genetic relations, neuronal networks, epidemiology of disease spread, diffusion of rumors, scientific knowledge networks, industrial collaborations, and international relations, to name a few. Despite this substantive diversity, researchers adopting a network perspective share a common conviction regarding the important consequences of interdependencies among units within their particular domains of interest. But beyond the focus on relations, the sheer omnivorousness of the network perspective means that it can, and does, admit a wide variety of applications.
Ian A. McCulloh, Helen L. Armstrong, and Anthony N. Johnson's book “Social Network Analysis with Applications” exemplifies the extraordinary potential of the network perspective with its specific focus on application of networks to organizational systems. In a crowded field of recent network books, McCulloh, Armstrong, and Johnson's stands out in providing an accessible introduction to network analysis, including substantive examples and interpretations grounded in organizational theories. Its practical applications to empirical data along with step-by-step instructions for analyzing networks using the Organizational Analyzer (ORA) software make it especially appropriate for newcomers to the study of networks.
One of the hallmarks of a maturing field is the translation of its core concepts and techniques into language that is readily accessible to newcomers and to practitioners who are primarily concerned with actionable results. McCulloh, Armstrong, and Johnson succeed admirably in that regard by drawing on their extensive combined experience teaching networks to soldiers, police officers, industry professionals, and both undergraduate and graduate students. Yet, Social Network Analysis with Applications insists on maintaining the formal rigor required for serious network analysis and does not shy away from math and equations when necessary to convey a concept. The book's unifying theoretical perspective and interpretative examples center on organizational risk and intervention. This focus draws attention to the consequences of particular organizational forms and individual positions and leads to consideration of how network actions and interventions might affect desirable or undesirable outcomes. This focus will surely provoke readers to consider the vast potential for applying network analysis in real-world settings.
I invite you to explore this book, to learn from it, and to use its many insightful examples and analytical approaches to expand your network imagination!
KATHERINE FAUST
Department of Sociology and Institute for Mathematical Behavioral Sciences
University of California, Irvine
Preface
We are very pleased to present “Social Network Analysis with Applications.” This book will focus on models and methods for social network analysis applied to organizational risk. Current books in the area of social network analysis are both highly technical and written at the advanced graduate level or they only discuss general concepts and omit mathematical calculations. Very few of the texts offer any practice problems for practitioners and students to complete as practice problems or homework exercises.
The inclusion of mathematical calculations is central to our approach in this text. Many in the field prefer to present social network analysis by hiding the mathematics and relying on computer software to identify centrality. We contend that this is a critical mistake in the pedagogy of network analysis. The authors have taught over 30 courses in social network analysis to over 500 students with varied approaches and consulted with many more colleagues. Those who have learned how to calculate centrality measures by hand calculation, for example, are 11 times more likely to retain an understanding of what the measures mean 1–3 months after the course. Thus, it is not our contention that an individual would use hand calculations on any real-world example. However, in learning to calculate measures by hand, the mathematics leads to an understanding of the underlying principles of social network analysis.
Many practitioners in industry, management, military intelligence, and law enforcement have expressed a growing interest in social network analysis, specifically focused on identifying organizational risk. We operationally define organizational risk as vulnerability in the social network. This could be a node high in informal power or a rare broker of resources. This could be a point of influence for the diffusion of ideology. There may exist many networks within an organization, such as a friendship network, a resource network, or a knowledge network. One or more of these networks may present organizational risk, while the others do not. In a military or law enforcement application, organizational risk identifies targets for further development and investigation. In an industry or management application, organizational risk identifies informal power brokers that should be included in management decisions, and potential vulnerability from lack of redundancy. The authors attempt to present examples of both.
The authors have trained soldiers in Iraq and Afghanistan, local and national police, and industry professionals on the applications of social network analysis. The topics laid out in this book follow the curriculum that they have used for the past 7 years in teaching week-long workshops as well as graduate and undergraduate level college courses.
The first three chapters introduce the mathematical concepts of a network and centrality. Again, we contend that an understanding of the mathematics leads to a deeper and more complete understanding of the social concepts behind social network analysis. Social network analysis software is also presented for larger problems. Visual analysis methods are also introduced. At the conclusion of the first three chapters, the reader should have a basic understanding of common network analysis techniques.
Chapters 5–7 provide the social theory that underlies social network analysis. It is infeasible to include all social theory related to social networks in the space afforded in this text. Therefore, the social theory that is most applicable to understanding organizational risk is provided. The authors' selection of the material included in these sections comes from experience applying social network analysis in industry, counter terrorism, and law enforcement. We have also sought input from former students who are actively using social network analysis on a regular basis. The reader is reminded that the focus of this text is for practitioners intending to apply social network analysis to organizations.
Chapters 7 and 8 are directed toward data. Matrix algebra is included in an appendix and provides a primer for the necessary mathematics to understand meta-networks and relational algebra. Relational algebra is an often overlooked method. While there is limited application for single mode networks, it is a critical tool for handling meta-networks, also known as multiplex or multimode networks. Relational algebra is the means to transform any relational data into different social networks, each of which might reveal organizational risk. Chapter 8 reviews sources of data. Strengths and weaknesses of data sources are presented. At the conclusion of Chapter 8, the reader should be capable of collecting network data and identifying organizational risk. The final chapter provides an organized approach to applying the methods and techniques presented in the book. It is intended to serve as a review, leading the reader through examples of applied social network analysis.
We are confident that you will enjoy “Social Network Analysis with Applications,” and feel empowered by the time you are finished. We have been amazed at the innovative and interesting ways our students have applied social network analysis to a wide variety of problems. More are sure to follow. Welcome to the fastest growing field in science!
IAN A. MCCULLOH
Curtin University, Australia
September, 2013
Acknowledgments
There are many people who have contributed to our understanding of social networks and the culmination of this text. For those interested, we present our journey of learning, while acknowledging those who made it possible.
In 2005, while conducting research in response surface optimization for the Army, I heard John Parmentola state, “Network Science is the Army's number four research priority.” As I began to learn more about this exciting field, I discovered several presentations from Steve Borgatti posted online. Steve is an amazing teacher and has the ability to present the most complex material in a very simple way. He was very open and collaborative with me and allowed me to use his posted materials when I taught my first social network analysis course at West Point in 2005.
Two students emerged out of my initial entry into social network analysis, Julie Paynter and Victor Basher. Had it not been for these two students, I probably would not have continued into the discipline. Julie was ranked number three in the class of 2006 at West Point. She was going to be a military intelligence officer and was majoring in mathematics. She had done an internship the summer before studying influence in jihadist texts. Her senior thesis applied social network analysis to understand influence in jihadist authors in Iraq. Vic Basher had been an intelligence analyst with the 10th Special Forces Group (Airborne) in Iraq and subsequently went to West Point. He was in my basic probability and statistics course, having difficulty understanding why an Army officer needed to learn math and physics. He was independently downloading insurgent propaganda videos on his personal computer to look for forensic clues that could help his friends who were at the time deployed to Iraq. In an effort to motivate his study of mathematics, I suggested an application of social network analysis to his data. As of April 1, 2006, Vic had downloaded 74% of all US deaths in Iraq and Afghanistan and had cross-referenced them with hometown news reports. The network we made allowed us to identify clusters (using one of Steve Borgatti's posted lectures) that helped us identify insurgent groups. Vic, Julie, and I presented our research at a variety of military audiences to include the Defense Intelligence Agency (DIA), National Security Agency (NSA), the newly formed Joint Improvised Explosive Device Defeat Organization (JIEDDO), the Asymmetric Warfare Group (AWG), and Special Operations Command (SOCOM).
During my tour of military agencies, I met Kathleen Carley. She told me that she was interested in collaborating with me. My brash response was that I “needed to get a Ph.D. out of the deal.” She put up with my arrogance. She taught me more than any professor or teacher I have ever known. She is one of the most brilliant academics I have ever met. Her contributions to the field of social network analysis are so advanced; many have difficulty believing they aren't science fiction. Few will ever know how many lives she saved through her contributions to the US Military's Global War on Terror. She showed the military how to target threats correctly, collect evidence, and prevent innocents from being erroneously targeted. I am very proud that she was willing to serve as myadvisor.
Around the same time, I received my first large research grant in the area of social networks. Joe Psotka and Dan Horn at the US Army Research Institute for the Behavioral and Social Sciences awarded me a grant to study and compare email networks with face-to-face friendship networks. In addition, they provided seed money to establish West Point's Network Science Center. As we stood up this center, my good friends Tony Johnson and Helen Armstrong took our efforts to a new level, expanding our research directions, organizing conferences, winning grants, and most importantly, teaching. Through our early conferences we met Nosh Contractor, László Barabási, and Guido Calderelli. Each of these individuals showed us different and novel applications of network analysis in social media, biology, physics, and fractals. They showed us how to reach out across our academic institution and create truly interdisciplinary and collaborative research.
Another student challenged my world-view of academia, Josh Lospinoso. I was the Mathematics Department academic advisor. Josh had recently majored in Operations Research and Tony and I couldn't understand why he had no interest in physics or engineering. He told us, “Those were easy problems that anyone could solve.” He was interested in the much more challenging and complex issues of human and social behavior; but not from a qualitative perspective. He wanted to model it mathematically. Josh joined Tony, Helen, and I as we learned about social networks together. Josh interned every summer at the NSA, applying social networks to defense applications as a cadet. Some argue that the success in Iraq in 2006–2007 had less to do with the surge of forces and more to do with the successful targeting of threats. Josh's work contributed to that effort.
We learned that Josh was not alone in his view that social science provided a challenging application area for mathematics. We discovered mathematical sociology and anthropology. Through the Sunbelt conferences we were able to meet Russ Bernard, Dimitrios Christopolous, Jeff Johnson, David Krackhardt, and many others who encouraged us in our research and motivated us to bring social network analysis to the undergraduate student populations we had at West Point. Tom Valente, in particular, had many conversations with us about teaching social network analysis. He not only shared his materials freely but helped us shape the way we present material and the content of our courses. He has even organized sessions at Sunbelt for professors to discuss the pedagogy of social network analysis.
Through the support and collaboration of all of these individuals, our program at West Point grew from Julie and Vic to seven undergraduate students presenting research at the 2007 Sunbelt conference in Corfu, Greece, to 23 students at the 2011 Sunbelt conference in Tampa, Florida. There are now well over a dozen faculty and 100 undergraduate students studying social network analysis at West Point. Helen established the Centre for Organizational Analysis in the School of Information Systems at Curtin University of Technology in Perth, Australia, which expands this effort and it continues to grow. Ian and Tony have worked to expand an understanding of social network analysisthroughout the US Military to include teaching 16 courses in Iraq and Afghanistan. We are grateful to the International Network of Social Network Analysts (INSNA) as an academic community, for being so receptive to us and enabling us to develop this text and encourage us in our academic endeavors.
I. A. M.
Introduction
The analysis of networks is not a recent trend. With historical examples dating back several centuries, its use in scientific enquiry has increased over the past few decades in particular. The term network can refer to any number of different types of networks, for example, social networks of people or a network of roads or a computer network. The Oxford dictionary defines a network as a group or system of interconnected people or things, suggesting that a network contains objects or people, and connections or links between these. By studying the objects, the links, and also the structure and dynamics of the network we can discover many important aspects that were previously not known. In this text, we focus on social network analysis. Linton Freeman describes the discipline.
“The social network approach is grounded in the intuitive notion that the patterning of social ties in which actors are embedded has important consequences for those actors. Network analysts, then, seek to uncover various kinds of patterns. And they try to determine the conditions under which those patterns arise and to discover their consequences” (Freeman, 2004).
People do not act in a manner independent from the context of their social interactions and environment. Moreover, there are patterns and behaviors of interaction that are common among people, if one can properly understand the social context in which they occur. This is not restricted to social ties between people. It may include knowledge, resources, tasks, beliefs, roles, organizations, and more, as well as peoples' interactions with these other factors. To better understand what social network analysis is, we provide Freeman's definition of four features that make up social network analysis:
“Beyond commitment to these four features, however, modern social network analysts also recognize that a wide range of empirical phenomena can be explored in terms of their structural patterning” (Freeman, 2004).
This structural perspective in the social sciences dates back to the establishment of the field. Comte (Martineau, 1895) identified two branches of sociology, statics, and dynamics. He further defined statics as the study of the “laws of social interconnection.” This is essentially the theoretical roots of social network analysis. The appeal of the network approach to problems has also appealed to others in a broad range of disciplines from physics to the other social sciences. As such, there are methods and tools that can be exchanged and applied to a wide variety of problems in many different fields. This has greatly enriched the development of the science and the contributions social network analysis has made to many different fields. Others in the early development of social science introduced data, methods, and theory, which we now recognize as social network analysis. Simmel (1909) defined society as existing “where a number of individuals enter into interaction.” The focus of the research of Simmel and his students was on the structure of social patterns.
Hobson, a famous economist at the turn of the twentieth century, presented an affiliation matrix on overlapping directorships of financiers in South Africa. There were five companies and six individuals. This was an early attempt at relational algebra, which we present in Chapter 7. This represents an important step forward in the ability to recognize and construct social network data. It also highlights how non-people nodes can impact our representation of a social network.
We find it interesting that the first evidence of social network data was derived from relational data rather than direct observation, which was not apparent in the literature until the 1920s. Systematic observation of social network data was introduced by Wellman (1926) who recorded data of preschool children during play. Blatz and Bott (1928) identified different forms of interaction among children. This enabled her to focus her observations on interaction specific to a research context.
The idea of different types of interaction, or different types of links, was the beginning of the formation of the underlying theory of meta-networks, proposed by Kathleen Carley years later. Meta-networks are discussed in Chapter 7. Recently, people have begun to refer to a subset of meta-networks as multilevel networks. In addition, Bott would focus on one child at a time to observe the different types of interaction with others. She recorded these data in matrices. Hagman compared the data collection approaches of observation and interview to reveal differences and bias in findings (Hagman, 1933). This became an important issue in social network analysis, studied extensively by Bernard and Killworth in a series of excellent papers on the topic. This issue is discussed in Chapter 8.
Most people credit the beginnings of social network analysis to Jacob Moreno with the graphical social network of interactions between school children in the New York Times (Moreno, 1932, 1934) displayed in Figure 1.1. Moreno initially referred to his research as “psychological geography,” but later changed the name to sociometry and started a journal of the same name in 1938.Moreno's work was largely theoretical and his social networks were more illustrative than mathematical representations. Recognizing this, Moreno teamed up with one of the earliest mathematical sociologists from Columbia University, Paul Lazarsfeld. Lazarsfeld developed a probabilistic model of a social network, which appeared in the first volume of Moreno's journal.
Figure 1.1 Emotions mapped by new geography.
Other groups began to explore social network analysis and structural approaches to social behavior. Lewin was a well-known sociologist who embraced the structural approach. He developed the Research Center for Group Dynamics at MIT with four of his former students including Cartwright and Festinger. Following Lewin's sudden death and MIT's decision to close the research center, the Center moved to the University of Michigan in 1948. Cartwright and Festinger recruited Frank Harary, who had earned his Ph.D. and was teaching in the Department of Mathematics, as a mathematical collaborator. Cartwright and Harary proposed a mathematical statement of the notion of cognitive balance proposed by Heider, which we discuss in Chapter 4. Others working in this research center conducted work on personal influence and the diffusion of rumors. In the 1950s, this group collected a famous multiyear data set on affinity in a university dormitory (Newcomb, 1961).
The Columbia University also made several important contributions to social network analysis. Robert Merton was a sociologist with a background in theory who teamed up with Paul Lazarsfeld, who was a mathematician working as a methodologist. They worked together throughout their careers, mainly in the areas of communication and what developed into market research. They also produced many Ph.D. students who became very influential in the field. Menzel and Katz (1956) investigated the diffusion of drug information among physicians, shaping our understanding of marketing. Peter Blau (1977) proposed homophily, which states that people with similar characteristics are more likely to meet. We present homophily in Chapter 4. Charles Kadushin developed the concept of social circles, which paved the way for subgroup analysis, which we discuss in Chapter 5.
Everett Rogers is another important figure in social network analysis during this time. His dissertation from Iowa State University involved some simple social network analysis methods in the study of the diffusion of innovations. He later introduced the concepts of early adopters and the bell curve and S-curves for the adoption of innovations. He extended his research to preventative health and other applications. As of the writing of this book, Rogers' book on “The Diffusion of Innovations” is the second most cited book in the social sciences. Noah Friedkin and Kathleen Carley extended Rogers' work to the diffusion of ideology, which is important for understanding radicalization. These concepts are presented in Chapter 6. Rogers also produced Ph.D. scholars who have made impressive contributions to the field, such as George Barnett, James Danowski, Peter Monge, Nan Lin, William Richards, and Thomas Valente.
In the 1960s, Davis and Leinhardt became interested in Cartwright and Harary's mathematical treatment of Heider's balance theory. Davis and Leinhardt introduced several formal mathematical models of relations between three actors, known as the triad. Their models dealt with transitivity, which is where actors develop relationships through their shared connection to others. Their work is presented in Chapter 4.
Independently, around the same time frame, mathematicians Paul Erds andAlfréd Rényi made revolutionary discoveries in the evolution of “random graphs.” For our purposes, a graph is synonymous with a network. Erds and Rényi use the term graph to refer to the field of mathematics called graph theory. This term was introduced by a chemist, Sylvester, in 1878, as mathematicians were applying their ideas to chemistry. In their eight papers, Erds and Rényi evaluate the properties of random graphs with n nodes and l links. For a random graph, G, containing no links, at each time step a randomly chosen link among the possible links is added to G. All of the possible links are equiprobable. A general model used to generate random graphs is as follows: “For a given p, , each potential link of G is chosen with probability p, independent of other links. Such a random graph is denoted by G(n,p), where each link is determined by flipping a coin, which has probability p of coming up heads.” In this model of random graphs, each link has an equal probability of occurring or not occurring within the graph. This random graph model also assumes that all nodes in the graph are present at the beginning and the number of nodes in the network is fixed and remains the same throughout the network's life. In addition, all nodes in this model are considered equal and are undistinguishable from each other.
Building upon Cartwright and Harary's work, and utilizing Erds' theory of random graphs as well as the class of uniform distributions associated with these graphs, Holland and Leinhardt (1981) developed a variety of statistical tests for the analysis of social networks. Using a uniform distribution, these tests spread the total probability mass equally over all possible outcomes, thereby giving an equal probability to the existence of a link between any two nodes in the network. These statistical tests were used to develop a reference frame or constant benchmark to which observed data could be compared in order to determine how “structured a particular network was, or how far the network deviated from the benchmark” (Wasserman and Faust, 1994). While this text does not address the statistical treatment of networks, it is important for interested readers to be aware that there are well-developed statistical tests for networks.
One of the most important figures in social network analysis was Harrison White. Harrison was a physicist who became interested in social behavior. He was working at Carnegie Tech, where he met Nobel laureate Herbert Simon. Herbert Simon was a founding father of many scientific fields to include artificial intelligence, organization theory, complex systems, and computer simulation. He was also the first to propose preferential attachment as a mechanism to explain power law distributions in the 1950s. Simon's influence on Harrison White led him to pursue a second Ph.D. in Sociology from Princeton. Harrison then moved to Harvard, where he produced some of the most important leaders in the field of social network analysis including Peter Bearman, Paul Bernard, Phillip Bonacich, Scott Boorman, Ronald Breiger, Kathleen Carley, Mark Granovetter, Joel Levine, and Barry Wellman.He introduced blockmodeling to understand subgroups and interactions between them, which we explore in Chapter 5. He proposed the concept of structural equivalence. These concepts are foundational to understanding network structure. Linton Freeman states, “Contemporary network analysis could never have emerged without Harrison White's contributions” (Freeman, 2004).
Linton Freeman is another prolific scholar in the field of social network analysis. He defined the network centrality measures that form the core metrics of social networks. These measures provide mathematical expressions of informal power, diffusion reach, and direct influence in a network. These concepts represent the foundation of modern social network analysis and are presented in Chapters 2 and 3. Freeman also produced many Ph.D.s who have made numerous important contributions to the field, including Stephen P. Borgatti, Katherine Faust, Sue C. Freeman, Jeffery Johnson, David Krackhardt, and Lee Sailer.
Equally important to the development of social network analysis was the growth of a community of scholars interested in the field. Their collaboration was facilitated greatly by H. Russell Bernard and Alvin Wolfe with their organization of the Sunbelt conferences. Initially, Russ received a grant while he was teaching at the West Virginia University to host a small meeting of mathematicians, sociologists, and anthropologists working in the area of social network analysis in 1974. Several follow-on meetings occurred. By 1980, Russ had moved to the University of Florida and developed a friendship with Alvin Wolfe at the University of South Florida who was also interested in social network analysis. They decided that Florida was a nice place and that people would want to visit, especially to discuss networks. So they organized the first Sunbelt Conferences in Tampa, Florida, in February of 1981 and 1982. They established three rules. There would be no sessions during peak tanning hours. The older guys would sponsor a hospitality social for the younger guys to facilitate interaction in an informal setting. The conferences would be held in a warm, enjoyable place during the winter to attract people to attend. The Sunbelt conferences have grown in size significantly over the years and they now hold sessions throughout the day in multiple parallel tracks; however, it remains one of the most collaborative and accessible conferences in academia, bringing scholars across many disciplines together to collaborate on social network analysis. Later, Barry Wellman formed the academic professional society known as the “International Network of Social Network Analysts” (INSNA), which now organizes Sunbelt as its official annual meeting as well as editing the journal Connections. The final major development in social network analysis was the development of computer software to enable the mathematical calculations of network structure. Linton Freeman teamed up with Martin Everett and Steve Borgatti to produce UCI Net, which is one of the most widely used social network analysis software programs. Pajek is a European software tool that is well suited for displaying large data sets. Kathleen Carley developed the Organizational Risk Analyzer (ORA), which is designed to analyze meta-networks with multiple types of nodes and links, as well as a variety of visualization and overtime analysis features. The practical labs in this text are demonstrated in ORA. Recently, an open-source statistics package, R, featured powerful network analysis tools to include exponential random graph modeling with the statnet package and actor-oriented modeling with the Rsiena package. There are of course many other social network analysis tools. Social network analysis software has certainly enhanced the capability and practical application of the field.
Social network analysis offers more than pictures. It provides an entirely new dimension of analysis for organizational behavior. Traditional analysis focuses on individual attributes. Social networks focus on relationships between individuals. Traditional analysis assumes statistical independence, where social network analysis focuses on dependent observations. Traditional analysis seeks to identify the correlation between significant factors and a response variable. Social network analysis seeks to identify organizational structure. The underlying mathematics behind traditional analysis is calculus, the language of change. The corresponding mathematics behind social network analysis is linear algebra and graph theory. These differences can be significant in terms of how someone looks at social dynamics. We hope you are empowered by this treatment of an exciting and powerful approach to social science.
References
Blatz, W. E. and Bott, H. M. (1928). Parents and the Pre-School Child. J. M. Dent.
Blau, P. M. (1977). Inequality and Heterogeneity: A Primitive Theory of Social Structure. Free Press.
Freeman, L. (2004). The Development of Social Network Analysis: A Study in the Sociology of Science. Empirical Press.
Hagman, E. (1933). The Companionships of Preschool Children, by Elizabeth Pleger Hagman,… [Foreword by George D. Stoddard.]. University.
Holland, P. and Leinhardt, S. (1981). An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association, 76(373):33–50.
Martineau, H. (1895). La Philosophie positive d'Auguste Comte. Luis Bahl.
Moreno, J. L. (1932). Application of the Group Method to Classification. National Committee on Prisons and Prison Labor.
Moreno, J. L. (1934). Who Shall Survive? Nervous and Mental Disease Publishing Company.
Newcomb, T. M. (1961). The Acquaintance Process. Holt, Rinehart and Winston.
Simmel, G. (1909). The problem of sociology. The American Journal of Sociology, 15(3):289–320.
Wasserman, S. and Faust, K. (1994). Social Network Analysis: Methods and Applications. Cambridge University Press.
Wellman, B. (1926). The Development of Motor Co-ordination in Young Children, an Experimental Study in the Control of Hand and Arm Movements, by Beth Wellman,… the University.
Part I
Network Basics
Chapter 1: What is a Network?
Injustice anywhere is a threat to justice everywhere. We are caught in an inescapable network of mutuality tied in a single garment of destiny. Whatever affects one directly affects all indirectly.
Martin Luther King Jr., Letter from a Birmingham jail, April 16,1963
In the movie, Good Will Hunting, the main character, Will Hunting, played by Matt Damon, is a janitor at a university. As he is mopping floors, he notices a problem posted on a board as a challenge to math students. The solution to the first two parts of the problem uses network analysis.
First Two Parts of the Good Will Hunting Problem
Let G be Figure 1.1 on the right.
Find the adjacency matrix
A
of the graph G.
Find the matrix giving the number of three-step walks for the graph G.
Figure 1.1 Graph G.
1.1 Basic Network Concepts
A network is a collection of points linked through some type of association. These points can represent any object or subject (e.g., people, places, and resources) and the links can represent any relationship between them (e.g., route, distance, family membership, and reporting structure). The network is graphically illustrated using lines, arcs, and symbols so the viewer can visualize and analyze the structure of the network more easily. A simple network of four points can be seen in Figure 1.1. In this network the points are people and the links are relationships between the people.
A graph is the visual representation of a set of points, frequently called vertices or nodes that are connected by line segments called edges or links. Social networks are graphs that contain a finite set or sets of actors which we call agents and the relation or relations defined between them. A social network would then be comprised of nodes representing people with the corresponding links representing the relationship between the people.
It is important to understand how to navigate through a network graph. The information gained can help us understand how information flow through a network can be used to analyze the placement of nodes within the network and gauge their significance. We will look at how to do this analysis in the next few chapters, but first we need to understand network graph navigation terminology. Figure 1.1 has four vertices or nodes. Each node is a person. Moving from one node to another along a single edge or link that joins them is a step. A walk is a series of steps from one node to another. The number of steps is the length of the walk. For instance, there is a walk of three steps from node 1 to node 3 using the steps 1 to 4, 4 to 2, and 2 to 3. A trail is a walk in which all the links are distinct, although some nodes may be included more than once. The length of a trail is the number of links it contains. For example, the length of the trail between nodes 3 and 4 is 2, where 3 to 2 is the first link, and 2 to 4 is the second link. A path is a walk in which all nodes and links are distinct. Note that every path is a trail and every trail is a walk. In application to social networks, we often focus on paths rather than trails or walks. An important property of a pair of nodes is whether or not there is a path between them. If there is a path between nodes and (say nodes 1 and node 4 in Figure 1.1), then the nodes are said to be reachable
