Percolation on Dense Graph Sequences

In this paper, we determine the percolation threshold for an arbitrary sequence of dense graphs (Gn). Let ¸n be the largest eigenvalue of the adjacency matrix of Gn, and let Gn(pn) be the random subgraph of Gn that is obtained by keeping each edge independently with probability pn. We show that the appearance of a giant component in Gn(pn) has a sharp threshold at pn = 1=¸n. In fact, we prove much more, that if (Gn) converges to an irreducible limit, then the density of the largest component of Gn(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, Lovasz, Sos and Vesztergombi. In addition to using basic properties of convergence, we make heavy use of the methods of Bollobas, Janson and Riordan, who used such branching processes to study the emergence of a giant component in a very broad family of sparse inhomogeneous random graphs.