Two Books on Random Graphs

Abstract

Problems concerning the structure of media have challenged mathematicians and physicists for many years, and the last twenty years have witnessed much progress. Probabilists and statistical physicists have developed and refined techniques for approaching quantitative as well as qualitative questions, such as the nature of phase transition in physical systems and the bulk properties thereof. The limited extent of finite-dimensional space (usually d = 2 or d = 3) has constrained progress. In a parallel development described in these two books, combinatorial theorists have developed an intricate theory of certain networks not subject to such constraints of dimensionality. Owing to the simplicity of definition of these so-called random graphs, rich and complex discoveries have been made about their inner structures. There are various different types of random graphs, of which the following is perhaps the most basic. Take n vertices labelled 1, 2, 3, . . ., n, and from the (n) available unordered pairs draw N at random; join these pairs with edges to obtain a graph wn. What are the properties of wn? In a paper which has since received much attention, Erd6s and R6nyi (1960) began to answer this question. They thought of such graphs as growing organisms, observing the properties of wn as n -s co when N = N(n) is a prescribed function of n. Some examples of their findings are as follows: