Networks, Crowds, and Markets

Over the past decade there has been a growing public fascination with the complex connect- edness of modern society. This connectedness is found in many incarnations: in the rapid growth of the Internet, in the ease with which global communication takes place, and in the ability of news and information as well as epidemics and financial crises to spread with surprising speed and intensity. These are phenomena that involve networks, incentives, and the aggregate behavior of groups of people; they are based on the links that connect us and the ways in which our decisions can have subtle consequences for others.

This introductory undergraduate textbook takes an interdisciplinary look at economics, sociology, computing and information science, and applied mathematics to understand net- works and behavior. It describes the emerging field of study that is growing at the interface of these areas, addressing fundamental questions about how the social, economic, and tech- nological worlds are connected.

David Easley is the Henry Scarborough Professor of Social Science and the Donald C. Opatrny ’74 Chair of the Department of Economics at Cornell University. He was previ- ously an Overseas Fellow of Churchill College, Cambridge. His research is in the fields of economics, finance, and decision theory. In economics, he focuses on learning, wealth dynamics, and natural selection in markets. In finance, his work focuses on market mi- crostructure and asset pricing. In decision theory, he works on modeling decision making in complex environments. He is a Fellow of the Econometric Society and is Chair of the NASDAQ-OMX Economic Advisory Board.

Jon Kleinberg is the Tisch University Professor in the Computer Science Department at Cornell University. He is a member of the National Academy of Engineering and the American Academy of Arts and Sciences. His research focuses on issues at the interface of networks and information, with an emphasis on the social and information networks that underpin the Web and other online media. He is the recipient of MacArthur, Packard, and Sloan Foundation Fellowships; the Nevanlinna Prize; the ACM-Infosys Foundation Award; and the National Academy of Sciences Award for Initiatives in Research.

Networks, Crowds, and Markets Reasoning about a Highly Connected World

David Easley Cornell University

Jon Kleinberg Cornell University

CAMBRIDGE UNIVERSITY PRESS

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Cambridge University Press

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ISBN-13 978-0-521-19533-1

ISBN-13 978-0-511-77675-5

© David Easley and Jon Kleinberg 2010

2010

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Contents

Preface page xi

1 Overview 1 1.1 Aspects of Networks 2 1.2 Central Themes and Topics 7

Part I Graph Theory and Social Networks

2 Graphs 21 2.1 Basic Definitions 21 2.2 Paths and Connectivity 23 2.3 Distance and Breadth-First Search 29 2.4 Network Data Sets: An Overview 35 2.5 Exercises 39

3 Strong and Weak Ties 43 3.1 Triadic Closure 44 3.2 The Strength of Weak Ties 46 3.3 Tie Strength and Network Structure in Large-Scale Data 51 3.4 Tie Strength, Social Media, and Passive Engagement 54 3.5 Closure, Structural Holes, and Social Capital 58 3.6 Advanced Material: Betweenness Measures and Graph Partitioning 62 3.7 Exercises 74

4 Networks in Their Surrounding Contexts 77 4.1 Homophily 77 4.2 Mechanisms Underlying Homophily: Selection and Social Influence 81 4.3 Affiliation 83 4.4 Tracking Link Formation in Online Data 88 4.5 A Spatial Model of Segregation 96 4.6 Exercises 103

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5 Positive and Negative Relationships 107 5.1 Structural Balance 107 5.2 Characterizing the Structure of Balanced Networks 110 5.3 Applications of Structural Balance 113 5.4 A Weaker Form of Structural Balance 115 5.5 Advanced Material: Generalizing the Definition of Structural Balance 118 5.6 Exercises 132

Part II Game Theory

6 Games 139 6.1 What Is a Game? 140 6.2 Reasoning about Behavior in a Game 142 6.3 Best Responses and Dominant Strategies 146 6.4 Nash Equilibrium 149 6.5 Multiple Equilibria: Coordination Games 151 6.6 Multiple Equilibria: The Hawk–Dove Game 154 6.7 Mixed Strategies 156 6.8 Mixed Strategies: Examples and Empirical Analysis 161 6.9 Pareto Optimality and Social Optimality 165

6.10 Advanced Material: Dominated Strategies and Dynamic Games 167 6.11 Exercises 179

7 Evolutionary Game Theory 189 7.1 Fitness as a Result of Interaction 190 7.2 Evolutionarily Stable Strategies 191 7.3 A General Description of Evolutionarily Stable Strategies 196 7.4 Relationship between Evolutionary and Nash Equilibria 197 7.5 Evolutionarily Stable Mixed Strategies 199 7.6 Exercises 204

8 Modeling Network Traffic Using Game Theory 207 8.1 Traffic at Equilibrium 207 8.2 Braess’s Paradox 209 8.3 Advanced Material: The Social Cost of Traffic at Equilibrium 211 8.4 Exercises 219

9 Auctions 225 9.1 Types of Auctions 225 9.2 When Are Auctions Appropriate? 226 9.3 Relationships between Different Auction Formats 228 9.4 Second-Price Auctions 229 9.5 First-Price Auctions and Other Formats 232 9.6 Common Values and the Winner’s Curse 233 9.7 Advanced Material: Bidding Strategies in First-Price and All-Pay

Auctions 234 9.8 Exercises 242

contents vii

Part III Markets and Strategic Interaction in Networks

10 Matching Markets 249 10.1 Bipartite Graphs and Perfect Matchings 249 10.2 Valuations and Optimal Assignments 253 10.3 Prices and the Market-Clearing Property 255 10.4 Constructing a Set of Market-Clearing Prices 258 10.5 How Does This Relate to Single-Item Auctions? 261 10.6 Advanced Material: A Proof of the Matching Theorem 262 10.7 Exercises 270

11 Network Models of Markets with Intermediaries 277 11.1 Price Setting in Markets 277 11.2 A Model of Trade on Networks 280 11.3 Equilibria in Trading Networks 286 11.4 Further Equilibrium Phenomena: Auctions and Ripple Effects 290 11.5 Social Welfare in Trading Networks 294 11.6 Trader Profits 295 11.7 Reflections on Trade with Intermediaries 297 11.8 Exercises 297

12 Bargaining and Power in Networks 301 12.1 Power in Social Networks 301 12.2 Experimental Studies of Power and Exchange 304 12.3 Results of Network Exchange Experiments 305 12.4 A Connection to Buyer–Seller Networks 309 12.5 Modeling Two-Person Interaction: The Nash Bargaining Solution 310 12.6 Modeling Two-Person Interaction: The Ultimatum Game 312 12.7 Modeling Network Exchange: Stable Outcomes 314 12.8 Modeling Network Exchange: Balanced Outcomes 317 12.9 Advanced Material: A Game-Theoretic Approach to Bargaining 320

12.10 Exercises 327

Part IV Information Networks and the World Wide Web

13 The Structure of the Web 333 13.1 The World Wide Web 333 13.2 Information Networks, Hypertext, and Associative Memory 335 13.3 The Web as a Directed Graph 340 13.4 The Bow-Tie Structure of the Web 344 13.5 The Emergence of Web 2.0 347 13.6 Exercises 349

14 Link Analysis and Web Search 351 14.1 Searching the Web: The Problem of Ranking 351 14.2 Link Analysis Using Hubs and Authorities 353 14.3 PageRank 358

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14.4 Applying Link Analysis in Modern Web Search 363 14.5 Applications beyond the Web 366 14.6 Advanced Material: Spectral Analysis, Random Walks, and Web

Search 368 14.7 Exercises 378

15 Sponsored Search Markets 385 15.1 Advertising Tied to Search Behavior 385 15.2 Advertising as a Matching Market 388 15.3 Encouraging Truthful Bidding in Matching Markets: The VCG

Principle 391 15.4 Analyzing the VCG Mechanism: Truth-Telling as a Dominant

Strategy 395 15.5 The Generalized Second-Price Auction 398 15.6 Equilibria of the Generalized Second-Price Auction 401 15.7 Ad Quality 404 15.8 Complex Queries and Interactions among Keywords 406 15.9 Advanced Material: VCG Prices and the Market-Clearing Property 407

15.10 Exercises 420

Part V Network Dynamics: Population Models

16 Information Cascades 425 16.1 Following the Crowd 425 16.2 A Simple Herding Experiment 427 16.3 Bayes’ Rule: A Model of Decision Making under Uncertainty 430 16.4 Bayes’ Rule in the Herding Experiment 434 16.5 A Simple, General Cascade Model 436 16.6 Sequential Decision Making and Cascades 440 16.7 Lessons from Cascades 442 16.8 Exercises 444

17 Network Effects 449 17.1 The Economy without Network Effects 450 17.2 The Economy with Network Effects 453 17.3 Stability, Instability, and Tipping Points 456 17.4 A Dynamic View of the Market 457 17.5 Industries with Network Goods 462 17.6 Mixing Individual Effects with Population-Level Effects 465 17.7 Advanced Material: Negative Externalities and the El Farol Bar

Problem 470 17.8 Exercises 476

18 Power Laws and Rich-Get-Richer Phenomena 479 18.1 Popularity as a Network Phenomenon 479 18.2 Power Laws 481 18.3 Rich-Get-Richer Models 482

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18.4 The Unpredictability of Rich-Get-Richer Effects 484 18.5 The Long Tail 486 18.6 The Effect of Search Tools and Recommendation Systems 489 18.7 Advanced Material: Analysis of Rich-Get-Richer Processes 490 18.8 Exercises 493

Part VI Network Dynamics: Structural Models

19 Cascading Behavior in Networks 497 19.1 Diffusion in Networks 497 19.2 Modeling Diffusion through a Network 499 19.3 Cascades and Clusters 506 19.4 Diffusion, Thresholds, and the Role of Weak Ties 509 19.5 Extensions of the Basic Cascade Model 512 19.6 Knowledge, Thresholds, and Collective Action 514 19.7 Advanced Material: The Cascade Capacity 517 19.8 Exercises 532

20 The Small-World Phenomenon 537 20.1 Six Degrees of Separation 537 20.2 Structure and Randomness 538 20.3 Decentralized Search 541 20.4 Modeling the Process of Decentralized Search 543 20.5 Empirical Analysis and Generalized Models 546 20.6 Core–Periphery Structures and Difficulties in Decentralized Search 552 20.7 Advanced Material: Analysis of Decentralized Search 554 20.8 Exercises 564

21 Epidemics 567 21.1 Diseases and the Networks That Transmit Them 567 21.2 Branching Processes 569 21.3 The SIR Epidemic Model 572 21.4 The SIS Epidemic Model 576 21.5 Synchronization 578 21.6 Transient Contacts and the Dangers of Concurrency 582 21.7 Genealogy, Genetic Inheritance, and Mitochondrial Eve 585 21.8 Advanced Material: Analysis of Branching and Coalescent Processes 590 21.9 Exercises 602

Part VII Institutions and Aggregate Behavior

22 Markets and Information 607 22.1 Markets with Exogenous Events 608 22.2 Horse Races, Betting, and Beliefs 609 22.3 Aggregate Beliefs and the “Wisdom of Crowds” 615 22.4 Prediction Markets and Stock Markets 618 22.5 Markets with Endogenous Events 622

x contents

22.6 The Market for Lemons 623 22.7 Asymmetric Information in Other Markets 627 22.8 Signaling Quality 631 22.9 Quality Uncertainty Online: Reputation Systems and Other

Mechanisms 632 22.10 Advanced Material: Wealth Dynamics in Markets 635 22.11 Exercises 641

23 Voting 645 23.1 Voting for Group Decision Making 645 23.2 Individual Preferences 646 23.3 Voting Systems: Majority Rule 649 23.4 Voting Systems: Positional Voting 654 23.5 Arrow’s Impossibility Theorem 657 23.6 Single-Peaked Preferences and the Median Voter Theorem 658 23.7 Voting as a Form of Information Aggregation 663 23.8 Insincere Voting for Information Aggregation 665 23.9 Jury Decisions and the Unanimity Rule 668

23.10 Sequential Voting and the Relation to Information Cascades 672 23.11 Advanced Material: A Proof of Arrow’s Impossibility Theorem 673 23.12 Exercises 678

24 Property Rights 681 24.1 Externalities and the Coase Theorem 681 24.2 The Tragedy of the Commons 685 24.3 Intellectual Property 688 24.4 Exercises 691

Bibliography 693 Index 711

Preface

Over the past decade, there has been a growing public fascination with the complex “connectedness” of modern society. This connectedness is found in many incarnations: in the rapid growth of the Internet and the Web, in the ease with which global communi- cation now takes place, and in the ability of news and information as well as epidemics and financial crises to spread around the world with surprising speed and intensity. These are phenomena that involve networks, incentives, and the aggregate behavior of groups of people; they are based on the links that connect us and the ways in which each of our decisions can have subtle consequences for the outcomes of everyone else.

Motivated by these developments in the world, there has been a coming-together of multiple scientific disciplines in an effort to understand how highly connected systems operate. Each discipline has contributed techniques and perspectives that are characteristically its own, and the resulting research effort exhibits an intriguing blend of these different flavors. From computer science and applied mathematics has come a framework for reasoning about how complexity arises, often unexpectedly, in systems that we design; from economics has come a perspective on how people’s behavior is affected by incentives and by their expectations about the behavior of others; and from sociology and the social sciences have come insights into the characteristic structures and interactions that arise within groups and populations. The resulting synthesis of ideas suggests the beginnings of a new area of study, focusing on the phenomena that take place within complex social, economic, and technological systems.

This book grew out of a course that we developed at Cornell, designed to introduce this topic and its underlying ideas to a broad student audience at an introductory level. The central concepts are fundamental and accessible ones, but they are dispersed across the research literatures of the many different fields contributing to the topic. The principal goal of this book is therefore to bring the essential ideas together in a single unified treatment and to present them in a way that requires as little background knowledge as possible.

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xii preface

Overview. The book is intended to be used at the introductory undergraduate level, and as such it has no formal prerequisites beyond a level of comfort with basic mathematical definitions at a precalculus level. In keeping with the introductory style, many of the ideas are developed in special cases and through illustrative examples; our goal is to take concepts and theories that are complex in their full generality and to provide simpler formulations where the essential ideas still come through.

In our use of the book, we find that many students are also interested in pursuing some of these topics more deeply, and so it is useful to provide pathways that lead from the introductory formulations into the more advanced literature on these topics. With this in mind, we provide optional sections labeled Advanced Material at the ends of most chapters. These advanced sections are qualitatively different from the other sections in the book; some draw on more advanced mathematics, and their presentation is at a more challenging level of conceptual complexity. Aside from the additional mathematical background required, however, even these advanced sections are self- contained; they are also strictly optional, in the sense that nothing elsewhere in the book depends on them.

Synopsis. The first chapter of the book provides a detailed description of the topics and issues that we cover. Here we give a briefer summary of the main focus areas.

The book is organized into seven parts of three to four chapters each. Parts I and II discuss the two main theories that underpin our investigations of networks and behavior: graph theory, which studies network structure, and game theory, which formulates models of behavior in environments where people’s decisions affect each other’s outcomes. Part III integrates these lines of thought into an analysis of the network structure of markets and the notion of power in such networks. Part IV pursues a different integration, discussing the World Wide Web as an information network, the problem of Web search, and the development of the markets that currently lie at the heart of the search industry. Parts V and VI study the dynamics of some of the fundamental processes that take place within networks and groups, including the ways in which people are influenced by the decisions of others. Part V pursues this topic at an aggregate scale, where we model interactions between an individual and the population as a whole. Part VI continues the analysis at the more fine-grained level of network structure, beginning with the question of influence and moving on to the dynamics of search processes and epidemics. Finally, Part VII considers how we can interpret fundamental social institutions – including markets, voting systems, and property rights – as mechanisms for productively shaping some of the phenomena we’ve been studying.

Use of the Book. The book is designed for teaching as well as for any reader who finds these topics interesting and wants to pursue them independently at a deeper level.

Several different types of courses can be taught from this book. When we teach from it at Cornell, the students in our class come from many different majors and have a wide variety of technical backgrounds; this diversity in the audience has served as our primary calibration in setting the introductory level of the book. Our course includes a portion of the material from each chapter; for the sake of concreteness, we provide the approximate weekly schedule we follow below. (There are three 50-minute lectures

preface xiii

each week, except that weeks 6 and 7 of our course contain only two lectures each. In each lecture, we don’t necessarily include all the details from each indicated section.)

Week 1: Chapters 1; 2.1–2.3; 3.1–3.3, 3.5; 4.1 Week 2: Chapters 5.1–5.3; 6.1–6.4; 6.5–6.9 Week 3: Chapters 8.1–8.2; 9.1–9.6; 10.1–10.2 Week 4: Chapters 10.3; 10.4–10.5; 11.1–11.2 Week 5: Chapters 11.3–11.4; 12.1–12.3; 12.5–12.6 Week 6: Chapters 12.7–12.8; 13 Week 7: Chapter 14.1–14.2; 14.3–14.4 Week 8: Chapter 15.1–15.2; 15.3–15.4; 15.5–15.6, 15.8 Week 9: Chapter 16.1–16.2; 16.3–16.4; 16.5–16.7 Week 10: Chapters 17.1–17.2; 17.3–17.5; 18 Week 11: Chapters 19.1–19.2; 19.3; 19.4, 19.6 Week 12: Chapter 22.1–22.4; 22.5–22.9; 7.1–7.4 Week 13: Chapters 20.1–20.2; 20.3–20.6; 21.1–21.5 Week 14: Chapters 23.1–23.5; 23.6–23.9; 24

There are many other paths that a course could follow through the book. First, a number of new courses are being developed at the interface of computer science and economics, focusing particularly on the role of economic reasoning in the design and behavior of modern computing systems. The book can be used for such courses in several ways, building on four chapters as a foundation: Chapter 2 on graphs, Chapter 6 on games, Chapter 9 on auctions, and Chapter 10 on matching markets. From here, a more expansive version of such a course could cover the remainder of Parts II and III, all of Parts IV and V, Chapter 19, and portions of Part VII. A more focused and potentially shorter version of such a course concerned principally with auctions, markets, and the online applications of these ideas could be constructed from Chapters 2, 6, 9, 10, 13, 15, 17, 18, and 22, and drawing on parts of Chapters 11, 12, 14, 16, and 19. When these courses are taught at a more advanced level, the advanced sections at the ends of most of these chapters would be appropriate material; depending on the exact level of the course, the text of many of these chapters could be used to lead into the more advanced analysis in their respective final sections.

In a different but related direction, new courses are also being developed on the topic of social computing and information networks. The book can be used for courses of this type by emphasizing Chapters 2–6, 13, 14, 17–20, and 22; many such courses also include sponsored search markets as part of their coverage of the Web, which can be done by including Chapters 9, 10, and 15 as well. The advanced sections in the book can play a role here too, depending on the level of the course.

Finally, portions of the book can serve as self-contained modules in courses on broader topics. To pick just a few examples, one can assemble such modules on network algorithms (Sections 2.3, 3.6, 5.5, 8.3, 10.6, 14.2, 14.3, 14.6, 15.9, 20.3, 20.4, and 20.7); applications of game theory (Chapters 6–9 and 11; Sections 12.9, 15.3– 15.6, 19.2, 19.3, 19.5–19.7, and 23.7–23.9); social network analysis (Chapters 2–5; Sections 12.1–12.3 and 12.5–12.8; and Chapters 18–20); the role of information in economic settings (Chapters 16 and 22, and Sections 23.6–23.10); and the analysis of large-scale network data sets (Sections 2.3, 3.2, 3.3, 3.6, 4.4, 5.3, 13.3, 13.4, 14.2–14.5, 18.2, 18.5, and 20.5). Most of these modules use graphs and/or games as fundamental

xiv preface

building blocks; for students not already familiar with these topics, Chapters 2 and 6, respectively, provide self-contained introductions.

Acknowledgments. Our work on this book took place in an environment at Cornell that was particularly conducive to interaction between computing and the social sciences. Our collaboration began as part of a project with Larry Blume, Eric Friedman, Joe Halpern, Dan Huttenlocher, and Éva Tardos funded by the National Science Foundation, followed by a campus-wide “theme project” on networks sponsored by Cornell’s Institute for the Social Sciences, with a group that included Larry and Dan together with John Abowd, Geri Gay, Michael Macy, Kathleen O’Connor, Jeff Prince, and David Strang. Our approach to the material in the book draws on perspectives – ways of thinking about these topics and ways of talking about them – that we’ve learned and acquired from this interdisciplinary set of colleagues, a group that includes some of our closest professional collaborators.

The course on which the book is based grew out of discussions that were part of the Cornell theme project; the two of us had taught distinct portions of this material separately in graduate courses that we had developed, and Michael Kearns’s Networked Life course at University of Pennsylvania demonstrated the vibrancy and relevance this material could have for an introductory undergraduate audience as well. We were intrigued by the prospect of combining different perspectives that hadn’t previously appeared together – a process that would be educational not only to the students in the course but to us as well. Creating and teaching this new interdisciplinary course was made possible by the support of our departments, Computer Science and Economics, and by support from the Solomon Fund at Cornell University.

Once the book had begun to take shape, we benefited enormously from the feed- back, suggestions, and experiences of colleagues who taught from early drafts of it. In particular, we thank Daron Acemoglu (MIT), Lada Adamic (Michigan), Allan Borodin (Toronto), Noshir Contractor (Northwestern), Jason Hartline (Northwestern), Nicole Immorlica (Northwestern), Ramesh Johari (Stanford), Samir Khuller (Mary- land), Jure Leskovec (Stanford), David Liben-Nowell (Carleton), Peter Monge (USC), Asu Ozdaglar (MIT), Vijay Ramachandran (Colgate), R. Ravi (CMU), Chuck Sev- erance (Michigan), Aravind Srinivasan (Maryland), and Luis von Ahn (CMU). The graduate and undergraduate teaching assistants in our own teaching of this subject have been very helpful as well; we thank Alex Ainslie, Lars Backstrom, Jacob Bank, Vlad Barash, Burak Bekdemir, Anand Bhaskar, Ben Cole, Bistra Dilkina, Eduard Dog- aru, Ram Dubey, Ethan Feldman, Ken Ferguson, Narie Foster, Eric Frackleton, Christie Gibson, Vaibhav Goel, Scott Grabnic, Jon Guarino, Fahad Karim, Koralai Kirabaeva, Tian Liang, Austin Lin, Fang Liu, Max Mihm, Sameer Nurmohamed, Ben Pu, Tal Rusak, Mark Sandler, Stuart Tettemer, Ozgur Yonter, Chong-Suk Yoon, and Yisong Yue.

In addition to the instructors who used early drafts, a number of other people provided extensive comments on portions of the book, leading to many improvements in the text: Lada Adamic, Robert Kerr, Evie Kleinberg, Gueorgi Kossinets, Stephen Morris, David Parkes, Rahul Sami, Andrew Tomkins, and Johan Ugander. We also thank a further set of colleagues, in addition to those already listed, who have provided very useful advice

preface xv

and suggestions on this project as it has proceeded: Bobby Kleinberg, Gene Kleinberg, Lillian Lee, Maureen O’Hara, Prabhakar Raghavan, and Steve Strogatz.

It has been a pleasure to be able to work with the editorial team at Cambridge University Press. Lauren Cowles, our main point of contact at Cambridge, has been an amazing source of advice and help, and we likewise very much appreciate the contributions of Scott Parris and David Tranah to this project, and Peggy Rote and her colleagues at Aptara for their work on the production of the book.

Finally, a profound thanks goes to our families, in continuing appreciation of their support and many other contributions.

David Easley Jon Kleinberg Ithaca, 2010

CHAPTER 1

Overview

The past decade has seen a growing public fascination with the complex “connected- ness” of modern society. At the heart of this fascination is the idea of a network – a pattern of interconnections among a set of things – and one finds networks appearing in discussion and commentary on an enormous range of topics. The diversity of con- texts in which networks are invoked is in fact so vast that it’s worth deferring precise definitions for a moment while we first recount a few of the more salient examples.

To begin with, the social networks we inhabit – the collections of social ties among friends – have grown steadily in complexity over the course of human history, due to technological advances facilitating distant travel, global communication, and digital interaction. The past half-century has seen these social networks depart even more radically from their geographic underpinnings – an effect that has weakened the tradi- tionally local nature of such structures but enriched them in other dimensions.

The information we consume has a similarly networked structure: these structures too have grown in complexity, as a landscape with a few purveyors of high-quality information (publishers, news organizations, the academy) has become crowded with an array of information sources of wildly varying perspectives, reliabilities, and motivating intentions. Understanding any one piece of information in this environment depends on understanding the way it is endorsed by and refers to other pieces of information within a large network of links.

Our technological and economic systems have also become dependent on networks of enormous complexity. This has made the behavior of these systems increasingly difficult to reason about and increasingly risky to tinker with. It has made them suscep- tible to disruptions that spread through the underlying network structures, sometimes turning localized breakdowns into cascading failures or financial crises.

The imagery of networks has made its way into many other lines of discussion as well: Global manufacturing operations now have networks of suppliers, Web sites have networks of users, and media companies have networks of advertisers. In such formulations, the emphasis is often less on the structure of the network itself than on its complexity as a large, diffuse population that reacts in unexpected ways to the actions of central authorities. The terminology of international conflict has come to reflect this

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Figure 1.1. The social network of friendships within a 34-person karate club [421]. (Drawing from the Journal of Anthropological Research.)

as well: for example, the picture of two opposing, state-supported armies gradually morphs, in U.S. presidential speeches, into images of a nation facing “a broad and adaptive terrorist network” [296] or “at war against a far-reaching network of violence and hatred” [328].

1.1 Aspects of Networks

How should we think about networks, at a more precise level, to bring all these issues together? In the most basic sense, a network is any collection of objects in which some pairs of these objects are connected by links. This definition is very flexible: depending on the setting, many different forms of relationships or connections can be used …

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