Sharp threshold for percolation on expanders

TL;DR

This paper proves that in large expander graphs with bounded degree, the emergence of a giant component in a random subgraph occurs abruptly at a specific probability threshold.

Contribution

It establishes a sharp threshold for the appearance of a giant component in random subgraphs of expanders with bounded degree.

Findings

Sharp threshold for giant component emergence in expanders

Threshold depends on edge probability p

Results apply to large finite graphs with bounded degree

Abstract

We study the appearance of the giant component in random subgraphs of a given large finite graph G=(V,E) in which each edge is present independently with probability p. We show that if G is an expander with vertices of bounded degree, then for any c in ]0,1[, the property that the random subgraph contains a giant component of size c|V| has a sharp threshold.