Percolation in General Graphs

We consider a random subgraph G(p) of a host graph G formed by retaining each edge of G with probability p. We address the question of determining the critical value p (as a function of G) for which a giant component emerges. Suppose G satisfies some (mild) conditions depending on its spectral gap and higher moments of its degree sequence. We define the second- order average degree (d) over tilde to be (d) over tilde = Sigma(v) d(v)(2) /(Sigma(v) d(v)), where d(v) denotes the degree of v. We prove that for any epsilon > 0, if p > (1 + epsilon)/(d) over tilde, then asymptotically almost surely, the percolated subgraph Gp has a giant component. In the other direction, if p < (1 - epsilon)/<(d)over tilde>, then almost surely, the percolated subgraph G(p) contains no giant component. An extended abstract of this paper appeared in the WAW 2009 proceedings [Chung et al. 09]. The main theorems are strengthened with much weaker assumptions.