Critical behavior in inhomogeneous random graphs

We study the critical behavior of inhomogeneous random graphs in the so-called rank-1 case, where edges are present independently but with unequal edge occupation probabilities. The edge occupation probabilities are moderated by vertex weights, and are such that the degree of vertex i is close in distribution to a Poisson random variable with parameter w(i), where w(i) denotes the weight of vertex i. We choose the weights such that the weight of a uniformly chosen vertex converges in distribution to a limiting random variable W. In this case, the proportion of vertices with degree k is close to the probability that a Poisson random variable with random parameter W takes the value k. We pay special attention to the power-law case, i. e., the case where P(W >= k) is proportional to k(-(tau-1)) for some power-law exponent tau > 3, a property which is then inherited by the asymptotic degree distribution. We show that the critical behavior depends sensitively on the properties of the asymptotic degree distribution moderated by the asymptotic weight distribution W. Indeed, when P(W > k) <= ck(-(tau-1)) for all k >= 1 and some tau > 4 and c > 0, the largest critical connected component in a graph of size n is of order n(2/3), as it is for the critical Erdos-Renyi random graph. When, instead, P(W > k) = ck(-(tau-1))(1 + o(1)) for k large and some tau is an element of (3, 4) and c > 0, the largest critical connected component is of the much smaller order n((tau-2)/(tau-1)). (C) 2012 Wiley Periodicals, Inc.