The critical activation density in graph bootstrap percolation

TL;DR

This paper investigates the critical activation density in graph bootstrap percolation across various graphs H, establishing thresholds and introducing the parameter (H) to understand activation efficiency and sharpness of percolation.

Contribution

It locates the percolation threshold p_c(n,H) for all graphs H, improving previous results and introducing the parameter (H) to analyze activation efficiency.

Findings

Determined the critical percolation threshold p_c(n,H) for every graph H.

Connected the threshold location to the critical density (H).

Raised open questions about the computability of (H).

Abstract

In graph bootstrap percolation, edges of an Erd\H{o}s-R\'enyi random graph ${\mathcal G}_{n,p}$ are initially active. Activation spreads to other edges of the complete graph $K_n$ by an iterative process governed by a fixed graph $H$, whereby an edge becomes active whenever it is the only inactive edge in a copy of $H$. If all edges of $K_n$ are eventually activated, we say the process $H$-percolates. The case $H=K_3$ corresponds to the classical sharp threshold for connectivity in ${\mathcal G}_{n,p}$. When $H=K_4$, there are close connections with $2$-neighbor bootstrap percolation from statistical physics. Varying $H$ produces a wide range of behaviors. In this work, for every graph $H$, we locate the critical $H$-percolation threshold $p_c(n,H)$, answering a question of Balogh, Bollob\'as, and Morris. Our general methods recover and improve several previous results. The location of $p_c(n,H)$ is related to a critical limiting density $\rho(H)$ of graphs that most efficiently activate a given edge. Introducing the parameter $\rho(H)$ raises several questions. For instance, it remains open whether $\rho(H)$ is computable in general, and its expression appears to indicate when the $H$-percolation threshold is sharp.