Anatomy of a Young Giant Component in the Random Graph

We provide a complete description of the giant component of the Erdos-Renyi random graph G(n, p) as soon as it emerges from the scaling window, i.e., for p = (1 + epsilon)/n where epsilon(3)n -> infinity and epsilon = o(1). Our description is particularly simple for epsilon = o(n(-1/4)), where the giant component C-1 is contiguous with the following model (i.e., every graph property that holds with high probability for this model also holds w.h.p. for C-1). Let Z be normal with mean 2/3 epsilon(3)n and variance epsilon(3)n, and let K be a random 3-regular graph on 2 [Z] vertices. Replace each edge of K by a path, where the path lengths are i.i.d. geometric with mean 1/epsilon. Finally, attach an independent Poisson(1 - epsilon)-Galton-Watson tree to each vertex. A similar picture is obtained for larger epsilon = o(1), in which case the random 3-regular graph is replaced by a random graph with N-k vertices of degree k for k >= 3, where N-k has mean and variance of order epsilon(k)n. This description enables us to determine fundamental characteristics of the supercritical random graph. Namely, we can infer the asymptotics of the diameter of the giant component for any rate of decay of epsilon, as well as the mixing time of the random walk on C-1. (C) 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 139-178, 2011