Small Worlds: The Dynamics of Networks between Order and Randomness

Small Worlds: The Dynamics of Networks between Order and Randomness

by Duncan J. Watts

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Product Details

ISBN-13: 9780691188331
Publisher: Princeton University Press
Publication date: 06/05/2018
Series: Princeton Studies in Complexity , #9
Sold by: Barnes & Noble
Format: NOOK Book
File size: 5 MB

About the Author

Duncan J. Watts, is Associate Professor of Sociology at Columbia University and an external faculty member of the Santa Fe Institute. He holds a Ph.D. in theoretical and applied mechanics from Cornell University and is the author of Six Degrees: The Science of A Connected Age. He lives in New York City.

Table of Contents

Prefacexiii
1Kevin Bacon, the Small World, and Why It All Matters3
Part IStructure9
2An Overview of the Small-World Phenomenon11
2.1Social Networks and the Small World11
2.1.1A Brief History of the Small World12
2.1.2Difficulties with the Real World20
2.1.3Reframing the Question to Consider All Worlds24
2.2Background on the Theory of Graphs25
2.2.1Basic Definitions25
2.2.2Length and Length Scaling27
2.2.3Neighbourhoods and Distribution Sequences31
2.2.4Clustering32
2.2.5"Lattice Graphs" and Random Graphs33
2.2.6Dimension and Embedding of Graphs39
2.2.7Alternative Definition of Clustering Coefficient40
3Big Worlds and Small Worlds: Models of Graphs41
3.1Relational Graphs42
3.1.1[alpha]-Graphs42
3.1.2A Stripped-Down Model: [beta]-Graphs66
3.1.3Shortcuts and Contractions: Model Invariance70
3.1.4Lies, Damned Lies, and (More) Statistics87
3.2Spatial Graphs91
3.2.1Uniform Spatial Graphs93
3.2.2Gaussian Spatial Graphs98
3.3Main Points in Review100
4Explanations and Ruminations101
4.1Going to Extremes101
4.1.1The Connected-Caveman World102
4.1.2Moore Graphs as Approximate Random Graphs109
4.2Transitions in Relational Graphs114
4.2.1Local and Global Length Scales114
4.2.2Length and Length Scaling116
4.2.3Clustering Coefficient117
4.2.4Contractions118
4.2.5Results and Comparisons with [beta]-Model120
4.3Transitions in Spatial Graphs127
4.3.1Spatial Length versus Graph Length127
4.3.2Length and Length Scaling128
4.3.3Clustering130
4.3.4Results and Comparisons132
4.4Variations on Spatial and Relational Graphs133
4.5Main Points in Review136
5"It's a Small World after All": Three Real Graphs138
5.1Making Bacon140
5.1.1Examining the Graph141
5.1.2Comparisons143
5.2The Power of Networks147
5.2.1Examining the System147
5.2.2Comparisons150
5.3A Worm's Eye View153
5.3.1Examining the System154
5.3.2Comparisons156
5.4Other Systems159
5.5Main Points in Review161
Part IIDynamics163
6The Spread of Infectious Disease in Structured Populations165
6.1A Brief Review of Disease Spreading166
6.2Analysis and Results168
6.2.1Introduction of the Problem168
6.2.2Permanent-Removal Dynamics169
6.2.3Temporary-Removal Dynamics176
6.3Main Points in Review180
7Global Computation in Cellular Automata181
7.1Background181
7.1.1Global Computation184
7.2Cellular Automata on Graphs187
7.2.1Density Classification187
7.2.2Synchronisation195
7.3Main Points in Review198
8Cooperation in a Small World: Games on Graphs199
8.1Background199
8.1.1The Prisoner's Dilemma200
8.1.2Spatial Prisoner's Dilemma204
8.1.3N-Player Prisoner's Dilemma206
8.1.4Evolution of Strategies207
8.2Emergence of Cooperation in a Homogeneous Population208
8.2.1Generalised Tit-for-Tat209
8.2.2Win-Stay, Lose-Shift214
8.3Evolution of Cooperation in a Heterogeneous Population219
8.4Main Points in Review221
9Global Synchrony in Populations of Coupled Phase Oscillators223
9.1Background223
9.2Kuramoto Oscillators on Graphs228
9.3Main Points in Review238
10Conclusions240
Notes243
Bibliography249
Index257

What People are Saying About This

Larry Blume

Theoretical research on social networks has been hampered by a lack of models which capture the essential properties of large numbers of graphs with only a few key parameters. All the dyads, triads and acyclic mappings which fill the social network literature lead merely to a long enumeration of special cases. The random graph models introduced by Watts provide a rich foundation for future analytical and empirical research. The applications to dynamics in part 2 illustrate the richness of these models and promise even more exciting work to come.
Larry Blume, Cornell University

J. Matthews

If you are a postgraduate looking to make your name or a seasoned researcher looking for new challenges, this book offers something rare: a chance to get in at the ground floor of a whole new area of research whose variety of exciting applications is exceeded only by their abundance.
Robert A. J. Matthews, Aston University, U.K.

J. J. Collins

Duncan Watts's and Steve Strogatz's 1998 Nature paper on 'The collective dynamics of small-world networks' reinvigorated interest in the small-world phenomenon. Now, in Small Worlds, Watts follows up on this work with a detailed but accessible account of small-world networks that will appeal to both scientists and nonscientists. With examples ranging from the Kevin Bacon Game to models for the spread of diseases, Watts provides a clear description of how the structure of small-world networks can be characterized and a sense of how the interconnectivity of such networks can lead to intriguing dynamics. Be sure to tell your friends and their friends about this book.

Levin

This is a remarkably novel analysis, with implications for a broad range of scientific disciplines, including neurobiology, sociology, ecology, economics, and epidemiology. . . . The results are potentially profoundly important.
Simon A. Levin, Department of Ecology and Evolutionary Biology, Princeton University

Marshall W. Meyer

Small Worlds is outstanding. Watts begins with a simple observation: clustered networks, networks characterized by a large fraction of short ties and a small fraction of 'shortcuts' linking clusters with one another, appear in diverse settings and more frequently than might be expected. Watts then demonstrates that the dynamical behavior of these networks is highly sensitive to structure. The book is must reading, although not easy reading, for social scientists interested in networks, decision-making, and organizational design.(In other words, this is a high-investment, high-payoff book.)

Collins

Duncan Watts's and Steve Strogatz's 1998 Nature paper on 'The collective dynamics of small-world networks' reinvigorated interest in the small-world phenomenon. Now, in Small Worlds, Watts follows up on this work with a detailed but accessible account of small-world networks that will appeal to both scientists and nonscientists. With examples ranging from the Kevin Bacon Game to models for the spread of diseases, Watts provides a clear description of how the structure of small-world networks can be characterized and a sense of how the interconnectivity of such networks can lead to intriguing dynamics. Be sure to tell your friends and their friends about this book.
J. J. Collins, Center for BioDynamics and Department of Biomedical Engineering, Boston University

Ditto

Duncan Watts has created that rarity of rarities: a book with enough fascinating facts and stories to keep the casual reader turning the pages coupled with enough engaging detail to satisfy the most technically sophisticated reader. Thus, whether you are just curious about the world around you or eager to begin your own small-world research, this is the definitive guide to the fascinating and profound world of small-world networks.
William L. Ditto, Applied Chaos Laboratory, Georgia Institute of Technology

Matthews

If you are a postgraduate looking to make your name or a seasoned researcher looking for new challenges, this book offers something rare: a chance to get in at the ground floor of a whole new area of research whose variety of exciting applications is exceeded only by their abundance.
Robert A. J. Matthews, Aston University, U.K.

Harrison White

Enchanting! A voyage of exploration with fascinating byroads that yet brings the reader to powerful and useable conclusions. This work is worthy of Stanley Milgram exactly because Watts goes well beyond the original visualization while retaining its transparency.
Harrison White, Department of Sociology, Columbia University

Gilbert Strang

A good book on a fascinating topic—why two widely separated people are often connected by a small number of steps from friend to friend. We do indeed live in a 'small world.' When something happens so often there must be a reason—Duncan Watts is looking for it.
Gilbert Strang, Department of Mathematics, Massachusetts Institute of Technology

Meyer

Small Worlds is outstanding. Watts begins with a simple observation: clustered networks, networks characterized by a large fraction of short ties and a small fraction of 'shortcuts' linking clusters with one another, appear in diverse settings and more frequently than might be expected. Watts then demonstrates that the dynamical behavior of these networks is highly sensitive to structure. The book is must reading, although not easy reading, for social scientists interested in networks, decision-making, and organizational design.(In other words, this is a high-investment, high-payoff book.)
Marshall W. Meyer, The Wharton School, University of Pennsylvania

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