Percolation in General Graphs

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 d̃ to be d̃ = ∑ v d 2 v/( ∑ v dv) where dv denotes the degree of v. We prove that for any > 0, if p > (1 + )/d̃ then asymptotically almost surely the percolated subgraph Gp has a giant component. In the other direction, if p < (1 − )/d̃ then almost surely the percolated subgraph Gp contains no giant component.