Universal behaviour of majority bootstrap percolation on high-dimensional geometric graphs

Abstract

Majority bootstrap percolation is a monotone cellular automaton that can be thought of as a model of infection spreading in networks. Starting with an initially infected set, new vertices become infected once at least half of their neighbours are infected. The average case behaviour of this process was studied on the $n$-dimensional hypercube by Balogh, Bollobás and Morris, who showed that there is a phase transition as the typical density of the initially infected set increases: For small enough densities the spread of infection is typically local, whereas for large enough densities typically the whole graph eventually becomes infected. Perhaps surprisingly, they showed that the critical window in which this phase transition occurs does not contain $p=1/2$, and they gave bounds on its width on a finer scale. In this paper we consider the majority bootstrap percolation process on a class of high-dimensional geometric graphs which includes many of the graph families on which percolation processes are typically considered, such as grids, tori and Hamming graphs, as well as other well-studied families of graphs such as (bipartite) Kneser graphs, including the odd graph and the middle layer graph. We show similar quantitative behaviour in terms of the location and width of the critical window for the majority bootstrap percolation process on this class of graphs.