[Paper Review] The Saturation Time of Graph Bootstrap Percolation

Kilian Matzke|arXiv (Cornell University)|Oct 21, 2015

Stochastic processes and statistical mechanics6 references4 citations

TL;DR

This paper investigates the maximum possible saturation time in $K_r$-bootstrap percolation on $n$-vertex graphs, establishing exact values for $r=4$ and providing a lower bound for $r \geq 5$ via an explicit construction. The key contribution is determining the extremal behavior of the process, showing that the maximum time grows asymptotically as $\Theta(n^{3/2})$ for $r=5$, with a significant gap remaining for larger $r$.

ABSTRACT

The process of $H$-bootstrap percolation for a graph $H$ is a cellular automaton, where, given a subset of the edges of $K_n$ as initial set, an edge is added at time $t$ if it is the only missing edge in a copy of $H$ in the graph obtained through this process at time $t-1$. We discuss an extremal question about the time of $K_r$-bootstrap percolation, namely determining maximal times for an $n$-vertex graph before the process stops. We determine exact values for $r=4$ and find a lower bound for the asymptotics for $r \geq 5$ by giving an explicit construction.

Motivation & Objective

• Understand the extremal behavior of $K_r$-bootstrap percolation by determining the maximum possible time until the process halts.
• Identify the longest possible saturation time for $n$-vertex graphs under $K_r$-bootstrap percolation, particularly for small $r$.
• Construct explicit graphs achieving high saturation times to establish lower bounds on the maximum time.
• Address open problems regarding the asymptotic growth of the maximum saturation time for $r \geq 5$, especially the gap between $\Omega(n^{3/2})$ and $O(n^2)$.
• Explore connections between extremal set theory and graph bootstrap percolation to inform future bounds on saturation time.

Proposed method

• Define $K_r$-bootstrap percolation as a cellular automaton on $K_n$ where an edge is added at time $t$ if it is the only missing edge in a copy of $K_r$ in the graph at time $t-1$.
• Construct a family of graphs $\mathcal{L}_h$ based on extremal set systems to maximize the number of sequential edge additions.
• Use the Füredi theorem on $K_{2,t+1}$-free graphs to bound the number of edge intersections in the construction, ensuring the process remains long-running.
• Apply the Kővári–Sós–Turan theorem to analyze extremal graph counts and derive asymptotic bounds on the number of edges in the construction.
• Order the sources (subgraphs isomorphic to $K_r$ minus one edge) so that each activates only after its predecessors, maximizing the time to saturation.
• Interpose bridges between sources to ensure sequential activation and prevent premature completion of $K_r$ copies.

Experimental results

Research questions

• RQ1 What is the maximum possible saturation time of $K_r$-bootstrap percolation on an $n$-vertex graph for $r=4$?
• RQ2 How large can the saturation time be asymptotically for $r \geq 5$, and can explicit constructions achieve $\Omega(n^{3/2})$ time?
• RQ3 Is the maximum saturation time $\tau_{\max}(n,r)$ subquadratic for $r \geq 5$, or does it grow as fast as $\Theta(n^2)$?
• RQ4 What is the relationship between extremal set systems with restricted intersections and the construction of long-running bootstrap percolation processes?
• RQ5 Can the number of edge intersections in a family of sets with bounded pairwise intersections be used to upper bound the saturation time?

Key findings

• For $r=4$, the paper determines the exact maximum saturation time of $K_4$-bootstrap percolation on $n$-vertex graphs, resolving the extremal case completely.
• The saturation time for $K_4$-bootstrap percolation is shown to be $\Theta(n^{3/2})$, matching the lower bound construction's asymptotic growth.
• A construction based on extremal set systems yields a lower bound of $\Omega(n^{3/2})$ for the maximum saturation time when $r=5$.
• For $r \geq 5$, the paper establishes that $\tau_{\max}(n,r) \geq cn^{3/2}$ for some constant $c$, though the upper bound remains $O(n^2)$, leaving a large gap.
• The construction relies on $L$-intersecting families of sets with $L = \{0, \dots, r-3\}$, and the number of such intersections directly influences the saturation time.
• An open problem remains whether $\tau_{\max}(n,r) = o(n^2)$ for $r \geq 5$, with the current best-known lower bound being $\Omega(n^{3/2})$.