Bootstrap Percolation, Connectivity, and Graph Distance

TL;DR

This paper studies bootstrap percolation on graphs, establishing conditions for minimal percolating sets, and exploring how graph connectivity and distance influence the percolation process and its speed.

Contribution

It provides a sufficient condition for size-2 percolating sets in diameter 2 graphs and links graph invariants to percolation dynamics.

Findings

Cardinality 2 percolating sets exist in diameter 2 graphs when r=2 under certain conditions.

Bounds on the number of rounds to percolation are derived based on graph distance invariants.

Connections between connectivity and bootstrap percolation are established.

Abstract

Bootstrap Percolation is a process defined on a graph which begins with an initial set of infected vertices. In each subsequent round, an uninfected vertex becomes infected if it is adjacent to at least $r$ previously infected vertices. If an initially infected set of vertices, $A_0$, begins a process in which every vertex of the graph eventually becomes infected, then we say that $A_0$ percolates. In this paper we investigate bootstrap percolation as it relates to graph distance and connectivity. We find a sufficient condition for the existence of cardinality 2 percolating sets in diameter 2 graphs when $r = 2$. We also investigate connections between connectivity and bootstrap percolation and lower and upper bounds on the number of rounds to percolation in terms of invariants related to graph distance.