Percolation on dense graph sequences

Abstract

In this paper we determine the percolation threshold for an arbitrary sequence of dense graphs (G_n). Let λ_n be the largest eigenvalue of the adjacency matrix of G_n, and let G_n(p_n) be the random subgraph of G_n obtained by keeping each edge independently with probability p_n. We show that the appearance of a giant component in G_n(p_n) has a sharp threshold at p_n=1/λ_n. In fact, we prove much more: if (G_n) converges to an irreducible limit, then the density of the largest component of G_n(c/n) tends to the survival probability of a multi-type branching process defined in terms of this limit. Here the notions of convergence and limit are those of Borgs, Chayes, Lovász, Sós and Vesztergombi.

In addition to using basic properties of convergence, we make heavy use of the methods of Bollobás, Janson and Riordan, who used multi-type branching processes to study the emergence of a giant component in a very broad family of sparse inhomogeneous random graphs.