The Giant Component in a Random Subgraph of a Given Graph

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 ˜d to be ˜d = Èv d2 v/( Èv dv) where dv denotes the degree of v. We prove that for any > 0, if p > (1 + )/ ˜d then almost surely the percolated subgraph Gp has a giant component. In the other direction, if p < (1− )/ ˜d then almost surely the percolated subgraph Gp contains no giant component.

• Type: Conference
• Year: 2009
• Where: WAW
• Authors: Fan Chung Graham, Paul Horn, Linyuan Lu
• Keywords: Algorithms, Giant Component, Random Subgraph Gp, Subgraph Gp, WAW 2009