## Small Worlds: The Dynamics of Networks Between Order and RandomnessEveryone 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. |

### What people are saying - Write a 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 | |

### Other editions - View all

Small Worlds: The Dynamics of Networks Between Order and Randomness Duncan J. Watts No preview available - 1999 |