Percolation in General Graphs

Abstract We consider a random subgraph Gp 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 to be = Σ v d 2 v /(Σ v d v ), where d v denotes the degree of v. We prove that for any ∊ > 0, if p > (1 + ∊)/, then asymptotically almost surely, the percolated subgraph Gp has a giant component. In the other direction, if p > (1 − ∊)/, then almost surely, the percolated subgraph Gp 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.

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