Percolation in General Graphs

Abstract

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 $\overline{d}$ to be $\overline{d} = \sum_v d^2_v/(\sum_v d_v)$, where $d_v$ denotes the degree of $v$. We prove that for any $\epsilon > 0$, if $p > (1 + \epsilon)/\overline{d}$, then asymptotically almost surely, the percolated subgraph $G_p$ has a giant component. In the other direction, if $p < (1 − \epsilon)/\overline{d}$, then almost surely, the percolated subgraph $G_p$ contains no giant component. An extended abstract of this paper appeared in the WAW 2009 proceedings. The main theorems are strengthened with much weaker assumptions.