Graph bootstrap percolation -- a discovery of slowness

TL;DR

This paper explores the dynamics of graph bootstrap percolation, focusing on the maximum duration before the infection stabilizes, revealing diverse behaviors and open problems in combinatorics.

Contribution

We systematically studied the maximum running time of infection spread in graph bootstrap percolation and analyzed how different infection rules influence this duration.

Findings

The infection process can last arbitrarily long depending on the initial set and rule.

Connections to additive, extremal, and probabilistic combinatorics influence the process.

Open problems highlight the complexity and richness of the infection dynamics.

Abstract

Graph bootstrap percolation is a discrete-time process capturing the spread of a virus on the edges of $K_n$. Given an initial set $G\subseteq K_n$ of infected edges, the transmission of the virus is governed by a fixed graph $H$: in each round of the process any edge $e$ of $K_n$ that is the last uninfected edge in a copy of $H$ in $K_n$ gets infected as well. Once infected, edges remain infected forever. The process was introduced by Bollob\'as in 1968 in the context of weak saturation and has since inspired a vast array of beautiful mathematics. The main focus of this survey is the extremal question of how long the infection process can last before stabilising. We give an exposition of our recent systematic study of this maximum running time and the influence of the infection rule $H$. The topic turns out to possess a wide variety of interesting behaviour, with connections to additive, extremal and probabilistic combinatorics. Along the way we encounter a number of surprises and attractive open problems.