What I Tell You Three Times Is True: Bootstrap Percolation in Small Worlds

Abstract

A bootstrap percolation process on a graph G is an process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round each uninfected node which has at least r infected neighbours becomes infected and remains so forever. The parameter r ≥ 2 is fixed. We analyse this process in the case where the underlying graph is an inhomogeneous random graph, which exhibits a power-law degree distribution, and initially there are a(n) randomly infected nodes. The main focus of this paper is the number of vertices that will have been infected by the end of the process. The main result of this work is that if the degree sequence of the random graph follows a power law with exponent β, where 2<β<3, then a sublinear number of initially infected vertices is enough to spread the infection over a linear fraction of the nodes of the random graph, with high probability. More specifically, we determine explicitly a critical function ac(n) such that ac(n)=o(n) with the following property. Assuming that n is the number of vertices of the underlying random graph, if a(n)≪ac(n), then the process does not evolve at all, with high probability as n grows, whereas if a(n)≫ac(n), then there is a constant e>0 such that, with high probability, the final set of infected vertices has size at least en. This behaviour is in sharp contrast with the case where the underlying graph is a G(n,p) random graph with p=d/n. Recent results of Janson, Łuczak, Turova and Vallier have shown that if the number of initially infected vertices is sublinear, then with high probability the size of the final set of infected vertices is approximately equal to a(n). That is, essentially there is lack of evolution of the process. It turns out that when the maximum degree is o(n1/(β−1)), then ac(n) depends also on r. But when the maximum degree is Θ(n1/(β−1)), then $a_c (n)=n^{\beta -2 \over \beta -1}$.