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Small Worlds : The Dynamics of Networks between Order and Randomness - Duncan J. Watts

Small Worlds

The Dynamics of Networks between Order and Randomness

Paperback Published: 24th November 2003
ISBN: 9780691117041
Number Of Pages: 280

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Everyone knows the small-world phenomenon: soon after meeting a stranger, we are surprised to discover that we have a mutual friend, or we are connected through a short chain of acquaintances. In his book, Duncan Watts uses this intriguing phenomenon--colloquially called "six degrees of separation"--as a prelude to a more general exploration: under what conditions can a small world arise in any kind of network?

The networks of this story are everywhere: the brain is a network of neurons; organisations are people networks; the global economy is a network of national economies, which are networks of markets, which are in turn networks of interacting producers and consumers. Food webs, ecosystems, and the Internet can all be represented as networks, as can strategies for solving a problem, topics in a conversation, and even words in a language. Many of these networks, the author claims, will turn out to be small worlds.

How do such networks matter? Simply put, local actions can have global consequences, and the relationship between local and global dynamics depends critically on the network's structure. Watts illustrates the subtleties of this relationship using a variety of simple models---the spread of infectious disease through a structured population; the evolution of cooperation in game theory; the computational capacity of cellular automata; and the sychronisation of coupled phase-oscillators.

Watts's novel approach is relevant to many problems that deal with network connectivity and complex systems' behaviour in general: How do diseases (or rumours) spread through social networks? How does cooperation evolve in large groups? How do cascading failures propagate through large power grids, or financial systems? What is the most efficient architecture for an organisation, or for a communications network? This fascinating exploration will be fruitful in a remarkable variety of fields, including physics and mathematics, as well as sociology, economics, and biology.

"An engaging and informative introduction."--Science "Playfully and clearly written... [Watts] uses examples adroitly, and mixes abstract theory with real-world anecdotes with superb skill... I have not enjoyed reading a book this much in a long time."--Peter Kareiva, Quarterly Review of Biology "[Small Worlds] will be seized on by those seeking a first rough map of this fascinating new mathematical land. Those entering can expect to find some amazing connections between areas of research with apparently nothing in common, such as neurology to business studies. But then, it's a small world."--Robert Matthews, New Scientist "Informally written and aimed at a wide audience, this book shows how mathematics yields new vistas on ubiquitous and seemingly familiar aspects of our world."--Choice

Prefacep. xiii
Kevin Bacon, the Small World, and Why It All Mattersp. 3
Structurep. 9
An Overview of the Small-World Phenomenonp. 11
Social Networks and the Small Worldp. 11
A Brief History of the Small Worldp. 12
Difficulties with the Real Worldp. 20
Reframing the Question to Consider All Worldsp. 24
Background on the Theory of Graphsp. 25
Basic Definitionsp. 25
Length and Length Scalingp. 27
Neighbourhoods and Distribution Sequencesp. 31
Clusteringp. 32
"Lattice Graphs" and Random Graphsp. 33
Dimension and Embedding of Graphsp. 39
Alternative Definition of Clustering Coefficientp. 40
Big Worlds and Small Worlds: Models of Graphsp. 41
Relational Graphsp. 42
[alpha]-Graphsp. 42
A Stripped-Down Model: [beta]-Graphsp. 66
Shortcuts and Contractions: Model Invariancep. 70
Lies, Damned Lies, and (More) Statisticsp. 87
Spatial Graphsp. 91
Uniform Spatial Graphsp. 93
Gaussian Spatial Graphsp. 98
Main Points in Reviewp. 100
Explanations and Ruminationsp. 101
Going to Extremesp. 101
The Connected-Caveman Worldp. 102
Moore Graphs as Approximate Random Graphsp. 109
Transitions in Relational Graphsp. 114
Local and Global Length Scalesp. 114
Length and Length Scalingp. 116
Clustering Coefficientp. 117
Contractionsp. 118
Results and Comparisons with [beta]-Modelp. 120
Transitions in Spatial Graphsp. 127
Spatial Length versus Graph Lengthp. 127
Length and Length Scalingp. 128
Clusteringp. 130
Results and Comparisonsp. 132
Variations on Spatial and Relational Graphsp. 133
Main Points in Reviewp. 136
"It's a Small World after All": Three Real Graphsp. 138
Making Baconp. 140
Examining the Graphp. 141
Comparisonsp. 143
The Power of Networksp. 147
Examining the Systemp. 147
Comparisonsp. 150
A Worm's Eye Viewp. 153
Examining the Systemp. 154
Comparisonsp. 156
Other Systemsp. 159
Main Points in Reviewp. 161
Dynamicsp. 163
The Spread of Infectious Disease in Structured Populationsp. 165
A Brief Review of Disease Spreadingp. 166
Analysis and Resultsp. 168
Introduction of the Problemp. 168
Permanent-Removal Dynamicsp. 169
Temporary-Removal Dynamicsp. 176
Main Points in Reviewp. 180
Global Computation in Cellular Automatap. 181
Backgroundp. 181
Global Computationp. 184
Cellular Automata on Graphsp. 187
Density Classificationp. 187
Synchronisationp. 195
Main Points in Reviewp. 198
Cooperation in a Small World: Games on Graphsp. 199
Backgroundp. 199
The Prisoner's Dilemmap. 200
Spatial Prisoner's Dilemmap. 204
N-Player Prisoner's Dilemmap. 206
Evolution of Strategiesp. 207
Emergence of Cooperation in a Homogeneous Populationp. 208
Generalised Tit-for-Tatp. 209
Win-Stay, Lose-Shiftp. 214
Evolution of Cooperation in a Heterogeneous Populationp. 219
Main Points in Reviewp. 221
Global Synchrony in Populations of Coupled Phase Oscillatorsp. 223
Backgroundp. 223
Kuramoto Oscillators on Graphsp. 228
Main Points in Reviewp. 238
Conclusionsp. 240
Notesp. 243
Bibliographyp. 249
Indexp. 257
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780691117041
ISBN-10: 0691117047
Series: Princeton Studies in Complexity
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 280
Published: 24th November 2003
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.4 x 15.6  x 1.5
Weight (kg): 0.43