Bootstrap percolation in random $k$-uniform hypergraphs

Abstract

We investigate bootstrap percolation with infection threshold $r>1$ on the binomial $k$-uniform random hypergraph $H_k(n,p)$ in the regime $n^{-1}\ll n^{k-2}p\ll n^{-1/r}$, when the initial set of infected vertices is chosen uniformly at random from all sets of given size. We establish a threshold such that if there are less vertices in the initial set of infected vertices, then whp only a few additional vertices become infected, while if the initial set of infected vertices exceeds the threshold then whp almost every vertex becomes infected. In addition, for $k=2$, we show that the probability of failure decreases exponentially.

Extended abstract presented at the European Conference on Combinatorics, Graph Theory and Applications 2015

Cited by in corpus (2)

• High-dimensional bootstrap processes in evolving simplicial complexes
• A large deviation approach to super-critical bootstrap percolation on the random graph $G_{n,p}$