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Graph bootstrap percolation -- a discovery of slowness

David Fabian, Patrick Morris, Tibor Szabó

TL;DR

This survey investigates graph bootstrap percolation on $K_n$, focusing on the maximum running time $M_H(n)$ and how the infection rule $H$ shapes it, linking to weak saturation and percolation thresholds. It develops chain-based constructions (including $K_k$-chains and ladder/dilation chains) and Behrend-type dilation sets to produce near-quadratic running times for many $H$, while isolating regimes (trees, cycles, bipartite rules) where the process stabilises in sub-linear or linear time. The authors present a comprehensive framework tying combinatorial constructions, probabilistic methods, and additive number theory to the dynamics of $H$-processes, and they highlight numerous open problems (notably the asymptotics for $M_{K_5}(n)$ and potential tree-width–based bounds). Overall, the work reveals rich, sometimes counterintuitive interactions between graph structure and bootstrap dynamics, offering both precise results and a broad agenda for future research in extremal and probabilistic combinatorics.

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 Bollobás 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.

Graph bootstrap percolation -- a discovery of slowness

TL;DR

This survey investigates graph bootstrap percolation on , focusing on the maximum running time and how the infection rule shapes it, linking to weak saturation and percolation thresholds. It develops chain-based constructions (including -chains and ladder/dilation chains) and Behrend-type dilation sets to produce near-quadratic running times for many , while isolating regimes (trees, cycles, bipartite rules) where the process stabilises in sub-linear or linear time. The authors present a comprehensive framework tying combinatorial constructions, probabilistic methods, and additive number theory to the dynamics of -processes, and they highlight numerous open problems (notably the asymptotics for and potential tree-width–based bounds). Overall, the work reveals rich, sometimes counterintuitive interactions between graph structure and bootstrap dynamics, offering both precise results and a broad agenda for future research in extremal and probabilistic combinatorics.

Abstract

Graph bootstrap percolation is a discrete-time process capturing the spread of a virus on the edges of . Given an initial set of infected edges, the transmission of the virus is governed by a fixed graph : in each round of the process any edge of that is the last uninfected edge in a copy of in gets infected as well. Once infected, edges remain infected forever. The process was introduced by Bollobás 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 . 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.
Paper Structure (42 sections, 31 theorems, 37 equations, 9 figures)

This paper contains 42 sections, 31 theorems, 37 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Weak saturation
  3. Percolation thresholds
  4. Maximum running times
  5. The content and the goal of this survey
  6. Organisation and notation
  7. What is not included
  8. Cliques and chains
  9. Weak saturation for cliques
  10. Percolation thresholds for cliques
  11. Maximum running times for cliques
  12. Proof highlight: $M_{K_5}(n)$ lower bound
  13. The construction
  14. The set of dilations.
  15. Finishing the proof
  16. ...and 27 more sections

Key Result

Lemma 2.3

Let $k\geq 5$ and $\tau\geq 1$ and suppose that $(H_i, e_i)_{i\in [\tau]}$ is a $K_k$-chain such that: Then if $G':=G-\{e_1,\ldots,e_\tau\}$ is the starting graph of the chain, we have that $\tau_H(G')\geq \tau$.

Figures (9)

  • Figure 1: A simple $K_4$-chain.
  • Figure 2: Sections of simple $K_6$-chains with slopes 0 (left), 4 (middle) and 8 (right). The edges $e_i$ of the simple chains are bold.
  • Figure 3: A linking $K_5$-chain $(L_i^j,g_i^j)_{i\in [3]}$ with the edges $g_i^j$ as dashed lines.
  • Figure 4: A starting graph maximising the running time of the $C_{k}$-process for $k\in 2\mathbb{N}$.
  • Figure 5: A graph $H$ with minimum degree $1$ and quadratic running time.
  • ...and 4 more figures

Theorems & Definitions (62)

  • Example 1.1: The triangle
  • Definition 2.1: A simple $K_k$-chain
  • Definition 2.2: $H$-chains
  • Lemma 2.3: A lower bound on running times for chains
  • proof
  • Conjecture 2.5
  • proof
  • Theorem 3.2: Behrend behrend1946sets
  • Corollary 3.3
  • Lemma 3.4
  • ...and 52 more