The Giant Component in a Random Subgraph of a Given Graph

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 $\tilde{d}$ to be $\tilde{d}=\sum_v d_v^2/(\sum_v d_v)$ where d _v denotes the degree of v. We prove that for any ε> 0, if $p > (1+ \epsilon)/{\tilde d}$ then asymptotically almost surely the percolated subgraph G _p has a giant component. In the other direction, if $p < (1-\epsilon)/\tilde{d}$ then almost surely the percolated subgraph G _p contains no giant component.