In this paper we determine the percolation threshold for an arbitrary sequence of dense graphs (Gn). Let λn be the largest eigenvalue of the adjacency matrix of Gn, and let Gn(pn) be the random subgraph of Gn 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/λn. In fact, we prove much more: 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, 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.
"Percolation on dense graph sequences." Ann. Probab. 38 (1) 150 - 183, January 2010. https://doi.org/10.1214/09-AOP478