Random Graph Dynamics

By: Rick Durrett

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Paperback | 31 May 2010

At a Glance

Paperback
224 Pages

Edition Type
Revised

Dimensions(cm)
29.69 x 21.01 x 1.19

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The theory of random graphs began in the late 1950s in several papers by Erdos and Renyi. In the late twentieth century, the notion of six degrees of separation, meaning that any two people on the planet can be connected by a short chain of people who know each other, inspired Strogatz and Watts to define the small world random graph in which each site is connected to k close neighbors, but also has long-range connections. At a similar time, it was observed in human social and sexual networks and on the Internet that the number of neighbors of an individual or computer has a power law distribution. This inspired Barabasi and Albert to define the preferential attachment model, which has these properties. These two papers have led to an explosion of research. The purpose of this book is to use a wide variety of mathematical argument to obtain insights into the properties of these graphs. A unique feature is the interest in the dynamics of process taking place on the graph in addition to their geometric properties, such as connectedness and diameter.

Industry Reviews

'A very valuable addition to the growing field of random graphs, providing a systematic coverage of these novel models.' Michael Krivelevich, Mathematical Reviews
'The book is written in a friendly, chatty style, making it easy to read; I very much like that. In summary, Random Graph Dynamics is a nice contribution to the area of random graphs and a source of valuable insights.' Malwina J. Luczak, Journal of the American Statistical Association

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Preface	p. vii
Overview	p. 1
Introduction to the Introduction	p. 1
Erdös, Rényi, Molloy, and Reed	p. 3
Six Degrees, Small Worlds	p. 7
Power Laws, Preferential Attachment	p. 11
Epidemics and Percolation	p. 15
Potts Models and the Contact Process	p. 18
Random Walks and Voter Models	p. 20
CHKNS Model	p. 21
Erdös-Rényi Random Graphs	p. 27
Branching Processes	p. 27
Cluster Growth as an Epidemic	p. 34
Cluster Growth as a Random Walk	p. 37
Diameter of the Giant Component	p. 43
CLT for the Giant Component	p. 46
Combinatorial Approach	p. 50
Critical Regime	p. 56
Threshold for Connectivity 62
Fixed Degree Distributions	p. 70
Definitions and Heuristics	p. 70
Proof of Phase Transition	p. 75
Subcritical Estimates	p. 82
Distances: Finite Variance	p. 84
Epidemics	p. 85
Power Laws	p. 90
Barabási-Albert Model	p. 90
Related Models	p. 93
Martingales and Urns	p. 99
Scale-Free Trees	p. 105
Distances: Power Laws 2 < ß < 3	p. 110
Diameter: Barabási-Albert Model	p. 116
Percolation, Resilience	p. 121
SIS Epidemic	p. 125
Small Worlds	p. 132
Watts and Strogatz Model	p. 132
Path Lengths	p. 134
Epidemics	p. 140
Ising and Potts Models	p. 144
Contact Process	p. 148
Random Walks	p. 153
Spectral Gap	p. 153
Conductance	p. 156
Fixed Degree Distribution	p. 159
Preferential Attachment Graph	p. 164
Connected Erdös-Rényi Graphs	p. 169
Small Worlds	p. 171
Only Degrees 2 and 3	p. 175
Hitting Times	p. 177
Voter Models	p. 181
CHKNS Model	p. 187
Heuristic Arguments	p. 187
Proof of the Phase Transition	p. 190
Subcritical Estimates	p. 193
Kosterlitz-Thouless Transition	p. 197
Results at the Critical Value	p. 200
References	p. 203
Index	p. 211
Table of Contents provided by Ingram. All Rights Reserved.

Random Graph Dynamics

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

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