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 ᵭ to be ᵭ = ∑v d2v/(∑v dv), where dv denotes the degree of v. We prove that for any∈ > 0, if p > (1 +∈)/ ᵭ, then asymptotically almost surely, the percolated subgraph Gp has a giant component. In the other direction, if p