The Giant component in a Random Subgraph of a Given Graph

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 degreed to bed where dv denotes the degree of v. We prove that for any > 0, if p > (1 + )/d then asymptotically 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

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 degreed to bed = v

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Percolation on Finite Cayley Graphs

In this paper, we study percolation on finite Cayley graphs. A conjecture of Benjamini says that the critical percolation pc of such a graph can be bounded away from one, for any Cayley graph satisfying a certain diameter condition. We prove Benjamini’s conjecture for some special classes of groups. We also establish a reduction theorem, which allows us to build Cayley graphs for large groups without increasing pc