Small Worlds: The Dynamics of Networks Between Order and RandomnessEveryone knows the smallworld 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 phenomenoncolloquially 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 modelsthe 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 phaseoscillators. 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  GoodreadsA 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  GoodreadsA 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 
249  
257  