The notion of six degrees of separation - that any two people on the planet can be connected by a short chain of people - inspired Strogatz and Watts to define the small world random graph, where each site is connected to close neighbours, but also has long range connections. At about the same time, it was observed in human social networks and on the internet that the number of neighbours of an individual has a power law distribution. This inspired Barabasi and Albert to define the preferential attachment model, which has these properties. These two papers led to an explosion of research, but much was nonrigorous and relied on simulations. This book uses mathematical arguments to obtain insights into these graphs. A unique feature of this book is the interest in the dynamics of process taking place on the graphs in addition to their geometric properties, like correctness and diameter.
Contents
1. Overview; 2. Erdos-Renyi random graphs; 3. Fixed degree distributions; 4. Power laws; 5. Small worlds; 6. Random walks; 7. CHKNS model.
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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