Small Worlds: The Dynamics of Networks Between Order and Randomness

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Princeton University Press, 1999 - Mathematics - 262 pages
10 Reviews

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.

  

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Review: Small Worlds: The Dynamics of Networks Between Order and Randomness

User Review  - Peter Flom - Goodreads

A formal look at networks from one of the pioneers in the field. Quite a bit of math. Read full review

Review: Small Worlds: The Dynamics of Networks Between Order and Randomness

User Review  - Goodreads

A formal look at networks from one of the pioneers in the field. Quite a bit of math. Read full review

Contents

Kevin Bacon the Small World and Why It All Matters
3
Part I Structure
9
An Overview of the SmallWorld Phenomenon
11
211 A Brief History of the Small World
12
212 Difficulties with the Real World
20
213 Reframing the Question to Consider All Worlds
24
22 Background on the Theory of Graphs
25
222 Length and Length Scaling
27
512 Comparisons
143
52 The Power of Networks
147
522 Comparisons
150
53 A Worms Eye View
153
531 Examining the System
154
532 Comparisons
156
54 Other Systems
159
55 Main Points in Review
161

223 Neighbourhoods and Distribution Sequences
31
224 Clustering
32
225 Lattice Graphs and Random Graphs
33
226 Dimension and Embedding of Graphs
39
227 Alternative Definition of Clustering Coefficient
40
Big Worlds and Small Worlds Models of Graphs
41
31 RELATIONAL GRAPHS
42
βGraphs
66
Model Invariance
70
314 Lies Damned Lies and More Statistics
87
32 Spatial Graphs
91
321 Uniform Spatial Graphs
93
322 Gaussian Spatial Graphs
98
33 Main Points in Review
100
Explanations and Ruminations
101
411 The ConnectedCaveman World
102
412 Moore Graphs as Approximate Random Graphs
109
42 Transitions in Relational Graphs
114
422 Length and Length Scaling
116
423 Clustering Coefficient
117
424 Contractions
118
425 Results and Comparisons with βModel
120
43 Transitions in Spatial Graphs
127
432 Length and Length Scaling
128
433 Clustering
130
434 Results and Comparisons
132
44 Variations on Spatial and Relational Graphs
133
45 Main Points in Review
136
Its a Small World after All Three Real Graphs
138
51 Making Bacon
140
511 Examining the Graph
141
Part II Dynamics
163
The Spread of Infectious Disease in Structured Populations
165
61 A Brief Review of Disease in Structured Populations
166
62 Analysis and Results
168
622 PermanentRemoval Dynamics
169
623 TemporaryRemoval Dynamics
176
63 Main Points in Review
180
Global Computation in Cellular Automata
181
711 Global Computation
184
72 Cellular Automata on Graphs
187
722 Synchronisation
195
73 Main Points in Review
198
Cooperation in a Small World Games on Graphs
199
811 The Prisoners Dilemma
200
812 Spatial Prisoners Dilemma
204
813 NPlayer Prisoners Dilemma
206
814 Evolution of Strategies
207
82 Emergence of Cooperation in a Homogeneous Population
208
821 Generalised TitforTat
209
822 WinStay LoseShift
214
83 Evolution of Cooperation in a Heterogeneous Population
219
84 Main Points in Review
221
Global Synchrony in Populations of Coupled Phase Oscillators
223
92 Kuramoto Oscillators on Graphs
228
93 Main Points in Review
238
Conclusions
240
Notes
243
Bibliography
249
Index
257
Copyright

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About the author (1999)

Duncan J. Watts, who received his Ph.D. in theoretical and applied mechanics from Cornell University in 1997, is a postdoctoral Fellow at the Santa Fe Institute.

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