Two Phase Transitions in Two-way Bootstrap Percolation

Abstract

Consider a graph $G$ and an initial random configuration, where each node is black with probability $p$ and white otherwise, independently. In discrete-time rounds, each node becomes black if it has at least $r$ black neighbors and white otherwise. We prove that this basic process exhibits a threshold behavior with two phase transitions when the underlying graph is a $d$-dimensional torus and identify the threshold values.

References in corpus (2)

• Dynamic Monopolies in Reversible Bootstrap Percolation
• The time of bootstrap percolation in two dimensions

Cited by in corpus (2)

• Target Set in Threshold Models
• Tight Bounds on the Minimum Size of a Dynamic Monopoly