Distances in random graphs with finite variance degrees

In this paper we study a random graph with N nodes, where node j has degree D-j and {D-j}IN j., are i.i.d. with P(D-j <= x) = F(x). We assume that 1 - F(x) <= cx(-tau + 1) for some tau > 3 and some constant c > 0. This graph model is a variant of the so-called configuration model, and includes heavy tail degrees with finite variance. The minimal number of edges between two arbitrary connected nodes, also known as the graph distance or the hopcount, is investigated when N -> infinity. We prove that the graph distance grows like log(nu) N, when the base of the logarithm equals v = E[D-j(D-j - 1)]/E[Dj] > 1. This confirms the heuristic argument of Newman, Strogatz, and Watts [Phys Rev E 64 (2002), 026118, 1- 17]. In addition, the random fluctuations around this asymptotic mean log,, N are characterized and shown to be uniformly bounded. In particular, we show convergence in distribution of the centered graph distance along exponentially growing subsequences. (c) 2005 Wiley Periodicals, Inc.