The time of graph bootstrap percolation

TL;DR

This paper investigates the timing of graph bootstrap percolation, determining the critical probability for percolation within a specific time frame in random graphs, extending previous work on percolation thresholds.

Contribution

It provides a near-complete characterization of the critical probability for percolation by time t in G(n,p), for all t up to a logarithmic scale, advancing understanding of percolation dynamics.

Findings

Critical probability determined up to a logarithmic factor for all t within C log log n

Extension of percolation threshold results to multiple time scales

Enhanced understanding of the percolation process in random graphs

Abstract

Graph bootstrap percolation, introduced by Bollob\'as in 1968, is a cellular automaton defined as follows. Given a "small" graph $H$ and a "large" graph $G = G_0 \subseteq K_n$, in consecutive steps we obtain $G_{t+1}$ from $G_t$ by adding to it all new edges $e$ such that $G_t \cup e$ contains a new copy of $H$. We say that $G$ percolates if for some $t \geq 0$, we have $G_t = K_n$. For $H = K_r$, the question about the size of the smallest percolating graphs was independently answered by Alon, Frankl and Kalai in the 1980's. Recently, Balogh, Bollob\'as and Morris considered graph bootstrap percolation for $G = G(n,p)$ and studied the critical probability $p_c(n,K_r)$, for the event that the graph percolates with high probability. In this paper, using the same setup, we determine, up to a logarithmic factor, the critical probability for percolation by time $t$ for all $1 \leq t \leq C \log\log n$.