Our purpose in writing this book is to provide a gentle introduction to a subject that is enjoying a surge in interest. We believe that the subject is fascinating in its own right, but the increase in interest can be attributed to several factors. One factor is the realization that networks are “everywhere.” From social networks such as Facebook, the World Wide Web and the Internet to the complex interactions between proteins in the cells of our bodies, we face the challenge of understanding their structure and development. By and large natural networks grow in an unpredictable manner and this is often modeled by a random construction. Another factor is the realization by Computer Scientists that NP-hard problems are often easier to solve than their worst-case suggests and that an analysis of running times on random instances can be informative.

History

Random graphs were used by Erdős [274] to give a probabilistic construction of a graph with large girth and large chromatic number. It was only later that Erdős and Rényi began a systematic study of random graphs as objects of interest in their own right. Early on they defined the random graph Gn,m and founded the subject. Often neglected in this story is the contribution of Gilbert [367] who introduced the model Gn,p, but clearly the credit for getting the subject started goes to Erdős and Rényi. Their seminal series of papers [275], [277], [278], [279] and, in particular, [276] on the evolution of random graphs laid the groundwork for other mathematicians to become involved in studying properties of random graphs.

In the early eighties the subject was beginning to blossom and it received a boost from two sources. First was the publication of the landmark book of Béla Bollobás [130] on random graphs. Around the same time, the Discrete Mathematics group at Adam Mickiewicz University began a series of conferences in 1983. This series continues biennially to this day and is now a conference attracting more and more participants.

The next important event in the subject was the start of the journal Random Structures and Algorithms in 1990 followed by Combinatorics, Probability and Computing a few years later. These journals provided a dedicated outlet for work in the area and are flourishing today.