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.

This book presents a wide-ranging variety of mathematical argument to give insights into the small world model, the preferential attachment model and related random graphs. A unique feature is the focus on the dynamics of process taking place on the graphs in addition to geometric properties such as correctness and diameter.

The notion of six degrees of separation, meaning that any two people can be connected by a short chain of people who know each other, inspired Strogatz and Watts to define the small world random graph. At about the same time, Barabasi and Albert defined the preferential attachment model-based on the properties of 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 book presents a wide-ranging variety of mathematical arguments to give insights into these random graphs.

A wide-ranging, rigorous look at properties of small world, preferential attachment, and related random graphs.

Rick Durrett is Professor of Mathematics at Cornell University. He received his Ph.D. in Operations Research from Stanford in 1976. After nine years at UCLA, he moved to Cornell, where his research turned to applications of probability to ecology and, more recently, genetics. He has written more than 150 papers, six other books, and has 33 academic descendants.

1. Overview; 2. Erdos-Renyi random graphs; 3. Fixed degree distributions; 4. Power laws; 5. Small worlds; 6. Random walks; 7. CHKNS model.