Strict majority bootstrap percolation in the r-wheel

TL;DR

This paper analyzes the strict majority bootstrap percolation process on r-wheel graphs, establishing that the critical initial activation probability for percolation approaches 1/4 as the graph size grows.

Contribution

It introduces the r-wheel graph model and determines the critical probability for percolation, showing a phase transition at p=1/4.

Findings

Percolation occurs with high probability if p>1/4 for large r.

Percolation probability is bounded away from 1 if p<1/4.

Critical probability converges to 1/4 as r increases.

Abstract

In this paper we study the strict majority bootstrap percolation process on graphs. Vertices may be active or passive. Initially, active vertices are chosen independently with probability p. Each passive vertex becomes active if at least half of its neighbors are active (and thereafter never changes its state). If at the end of the process all vertices become active then we say that the initial set of active vertices percolates on the graph. We address the problem of finding graphs for which percolation is likely to occur for small values of p. Specifically, we study a graph that we call r-wheel: a ring of n vertices augmented with a universal vertex where each vertex in the ring is connected to its r closest neighbors to the left and to its r closest neighbors to the right. We prove that the critical probability is 1/4. In other words, if p>1/4 then for large values of r percolation occurs with probability arbitrarily close to 1 as n goes to infinity. On the other hand, if p<1/4 then the probability of percolation is bounded away from 1.