Contagious Sets in Dense Graphs

TL;DR

This paper investigates the conditions under which small contagious sets exist in dense graphs and analyzes the speed of activation spread in bootstrap percolation, providing tight bounds and exact thresholds.

Contribution

It establishes minimum degree conditions for the existence of smallest contagious sets and determines exact edge thresholds for certain parameters in dense graphs.

Findings

Contagious sets of size r exist if minimum degree ≥ (k-1)/k * n.

Tight bounds on activation rounds until full activation are provided.

Exact maximum edge counts for graphs with no small contagious sets are determined.

Abstract

We study the activation process in undirected graphs known as bootstrap percolation: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it had at least r active neighbors, for a threshold r that is identical for all vertices. A contagious set is a vertex set whose activation results with the entire graph being active. Let m(G,r) be the size of a smallest contagious set in a graph G on n vertices. We examine density conditions that ensure m(G,r) = r for all r >= 2. With respect to the minimum degree, we prove that such a smallest possible contagious set is guaranteed to exist if and only if G has minimum degree at least (k-1)/k * n. Moreover, we study the speed with which the activation spreads and provide tight upper bounds on the number of rounds it takes until all nodes are activated in such a graph. We also investigate what average degree asserts the existence of small contagious sets. For n >= k >= r, we denote by M(n,k,r) the maximum number of edges in an n-vertex graph G satisfying m(G,r)>k. We determine the precise value of M(n,k,2) and M(n,k,k), assuming that n is sufficiently large compared to k.