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Tables of Contents for Small Worlds
Chapter/Section Title
Page #
Page Count
PREFACE
xiii
 
1 Kevin Bacon, the Small World, and Why It All Matters
3
6
PART I STRUCTURE
9
154
2 An Overview of the Small-World Phenomenon
11
30
2.1 Social Networks and the Small World
11
14
2.1.1 A Brief History of the Small World
12
8
2.1.2 Difficulties with the Real World
20
4
2.1.3 Reframing the Question to Consider All Worlds
24
1
2.2 Background on the Theory of Graphs
25
16
2.2.1 Basic Definitions
25
2
2.2.2 Length and Length Scaling
27
4
2.2.3 Neighbourhoods and Distribution Sequences
31
1
2.2.4 Clustering
32
1
2.2.5 "Lattice Graphs" and Random Graphs
33
6
2.2.6 Dimension and Embedding of Graphs
39
2
3 Big Worlds and Small Worlds: Models of Graphs
41
60
3.1 Relational Graphs
42
49
3.1.1 Alpha-Graphs
42
24
3.1.2 A Stripped-Down Model: Beta-Graphs
66
4
3.1.3 Shortcuts and Contractions: Model Invariance
70
17
3.1.4 Lies, Damned Lies, and (More) Statistics
87
4
3.2 Spatial Graphs
91
9
3.2.1 Uniform Spatial Graphs
93
5
3.2.2 Gaussian Spatial Graphs
98
2
3.3 Main Points in Review
100
1
4 Explanations and Ruminations
101
37
4.1 Going to Extremes
101
13
4.1.1 The Connected-Caveman World
102
7
4.1.2 Moore Graphs as Approximate Random Graphs
109
5
4.2 Transitions in Relational Graphs
114
13
4.2.1 Local and Global Length Scales
114
2
4.2.2 Length and Length Scaling
116
1
4.2.3 Clustering Coefficient
117
1
4.2.4 Contractions
118
2
4.2.5 Results and Comparisons with Beta-Model
120
7
4.3 Transitions in Spatial Graphs
127
6
4.3.1 Spatial Length versus Graph Length
127
1
4.3.2 Length and Length Scaling
128
2
4.3.3 Clustering
130
2
4.3.4 Results and Comparisons
132
1
4.4 Variations on Spatial and Relational Graphs
133
3
4.5 Main Points in Review
136
2
5 "It's a Small World after All": Three Real Graphs
138
25
5.1 Making Bacon
140
7
5.1.1 Examining the Graph
141
2
5.1.2 Comparisons
143
4
5.2 The Power of Networks
147
6
5.2.1 Examining the System
147
3
5.2.2 Comparisons
150
3
5.3 A Worm's Eye View
153
6
5.3.1 Examining the System
154
2
5.3.2 Comparisons
156
3
5.4 Other Systems
159
2
5.5 Main Points in Review
161
2
PART II DYNAMICS
163
80
6 The Spread of Infectious Disease in Structured Populations
165
16
6.1 A Brief Review of Disease Spreading
166
2
6.2 Analysis and Results
168
12
6.2.1 Introduction of the Problem
168
1
6.2.2 Permanent-Removal Dynamics
169
7
6.2.3 Temporary-Removal Dynamics
176
4
6.3 Main Points in Review
180
1
7 Global Computation in Cellular Automata
181
18
7.1 Background
181
6
7.1.1 Global Computation
184
3
7.2 Cellular Automata on Graphs
187
11
7.2.1 Density Classification
187
8
7.2.2 Synchronisation
195
3
7.3 Main Points in Review
198
1
8 Cooperation in a Small World: Games on Graphs
199
24
8.1 Background
199
9
8.1.1 The Prisoner's Dilemma
200
4
8.1.2 Spatial Prisoner's Dilemma
204
2
8.1.3 N-Player Prisoner's Dilemma
206
1
8.1.4 Evolution of Strategies
207
1
8.2 Emergence of Cooperation in a Homogeneous Population
208
11
8.2.1 Generalised Tit-for-Tat
209
5
8.2.2 Win-Stay, Lose-Shift
214
5
8.3 Evolution of Cooperation in a Heterogeneous Population
219
2
8.4 Main Points in Review
221
2
9 Global Synchrony in Populations of Coupled Phase Oscillators
223
17
9.1 Background
223
5
9.2 Kuramoto Oscillators on Graphs
228
10
9.3 Main Points in Review
238
2
10 Conclusions
240
3
NOTES
243
6
BIBLIOGRAPHY
249
8
INDEX
257