Random Graph Dynamics (Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 20) by Rick Durrett (PDF)

Description

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 about the same 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. While this literature is extensive, many of the papers are based on simulations and nonrigorous arguments. 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 of this book is the interest in the dynamics of process taking place on the graph in addition to their geometric properties, such as connectedness and diameter.

User’s Reviews

Editorial Reviews: Review “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 Book Description This book uses mathematical arguments to obtain insights into random graphs.

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