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

TL;DR

This paper investigates the behavior of majority bootstrap percolation on high-dimensional geometric graphs, revealing phase transition phenomena similar to those observed in hypercubes, with implications for understanding infection spread in complex networks.

Contribution

It extends the analysis of majority bootstrap percolation to a broad class of high-dimensional geometric graphs, establishing phase transition properties and critical window bounds.

Findings

Identifies phase transition behavior on various high-dimensional graphs.

Shows the critical window is bounded away from 1/2.

Provides bounds on the width of the critical window.

Abstract

Majority bootstrap percolation is a monotone cellular automata that can be thought of as a model of infection spreading in networks. Starting with an initially infected set, new vertices become infected once more than half of their neighbours are infected. The average case behaviour of this process was studied on the $n$-dimensional hypercube by Balogh, Bollob\'{a}s 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 is bounded away from $1/2$, 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.