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Universal behaviour of majority bootstrap percolation on high-dimensional geometric graphs

Maurício Collares, Joshua Erde, Anna Geisler, Mihyun Kang

Abstract

Majority bootstrap percolation is a monotone cellular automata that can be thought of as a model of infection spreading in networks. Starting with an initially infected set, new vertices become infected once more than half of their neighbours are infected. The average case behaviour of this process was studied on the $n$-dimensional hypercube by Balogh, Bollobás and Morris, who showed that there is a phase transition as the typical density of the initially infected set increases: For small enough densities the spread of infection is typically local, whereas for large enough densities typically the whole graph eventually becomes infected. Perhaps surprisingly, they showed that the critical window in which this phase transition occurs is bounded away from $1/2$, and they gave bounds on its width on a finer scale. In this paper we consider the majority bootstrap percolation process on a class of high-dimensional geometric graphs which includes many of the graph families on which percolation processes are typically considered, such as grids, tori and Hamming graphs, as well as other well-studied families of graphs such as (bipartite) Kneser graphs, including the odd graph and the middle layer graph. We show similar quantitative behaviour in terms of the location and width of the critical window for the majority bootstrap percolation process on this class of graphs.

Universal behaviour of majority bootstrap percolation on high-dimensional geometric graphs

Abstract

Majority bootstrap percolation is a monotone cellular automata that can be thought of as a model of infection spreading in networks. Starting with an initially infected set, new vertices become infected once more than half of their neighbours are infected. The average case behaviour of this process was studied on the -dimensional hypercube by Balogh, Bollobás and Morris, who showed that there is a phase transition as the typical density of the initially infected set increases: For small enough densities the spread of infection is typically local, whereas for large enough densities typically the whole graph eventually becomes infected. Perhaps surprisingly, they showed that the critical window in which this phase transition occurs is bounded away from , and they gave bounds on its width on a finer scale. In this paper we consider the majority bootstrap percolation process on a class of high-dimensional geometric graphs which includes many of the graph families on which percolation processes are typically considered, such as grids, tori and Hamming graphs, as well as other well-studied families of graphs such as (bipartite) Kneser graphs, including the odd graph and the middle layer graph. We show similar quantitative behaviour in terms of the location and width of the critical window for the majority bootstrap percolation process on this class of graphs.

Paper Structure

This paper contains 21 sections, 23 theorems, 165 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main Results
  4. Proof Techniques
  5. Preliminaries
  6. Probabilistic tools
  7. Nice structural properties of geometric graphs
  8. Proof of the 1-statement of Theorem \ref{['t:mainThm']}
  9. Proof of the 0-statement of Theorem \ref{['t:mainThm']}
  10. Eleven rounds suffice
  11. Proof of Lemma \ref{['l:constant']}
  12. Proof of Lemma \ref{['l:dimred']}
  13. Proof of Lemma \ref{['l:super-exponential']}
  14. Dominating process: stabilisation after two rounds
  15. Proof of Lemma \ref{['l:inA3-A2']}
  16. ...and 6 more sections

Key Result

Theorem \oldthetheorem

Let $d \geqslant r \geqslant 1$ and consider the random $(r+1)$-neighbour bootstrap percolation process. Then where $\lambda(d, r)$ is an explicit constant depending on $d$ and $r$, and $\log_r$ is the iterated logarithm.

Figures (4)

  • Figure 1: The set $X_0 =N(x) \cap A_0$ is depicted in blue, and the set $X_1=N(x) \cap (A_1 \setminus A_0)$ is depicted patterned. The red pair of vertices has two common neighbours in $S(x, 2)$ and is thus in $M'$, while the yellow pair of vertices has just one common neighbour in $S(x, 2)$.
  • Figure 2: The set $T' \subseteq N(x) \setminus A_4$ is depicted patterned. We will estimate the density of certain types of edges between $T= N(T') \cap S(x, 2)$ and $S_0(x, 3)$, where $D\cap S(x,3)$ is depicted in grey. Applying \ref{['l:partitiondist']}, we obtain a partition $S_0(x, 3)$ into sets of vertices with pairwise distance at least $6$.
  • Figure 3: For two vertices $y_1, y_2 \in S_0(x, 3)$, by applying \ref{['c:projection']} there exist subgraphs $G(y_1)$ and $G(y_2)$ that are disjoint from $B(x, 2)$ and lie in $\mathcal{H}(K)$.
  • Figure 4: For a vertex $x \in (\hat{A}_3 \setminus \hat{A}_2)$ there is a set $W_1 \subseteq S_0(x,1) \cap (\hat{A}_2 \setminus \hat{A}_1)$ of size $\gamma-3K$. For each $w \in W_1$ there is a set $W_{2, w} \subseteq S_0(x, 2) \cap \hat{A}_1 \setminus \hat{A}_0$, and their union is $W_2'$, which contains a subset $W_2$ of size exactly $(\gamma-3K)^2/2K$.

Theorems & Definitions (67)

  • Theorem \oldthetheorem: balogh2011Grid
  • Theorem \oldthetheorem: BaBoMo2009, informal
  • Theorem \oldthetheorem: BaBoMo2009
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem: BaBoMo2009, Lemmas 3.8 and 3.9
  • Proposition \oldthetheorem
  • ...and 57 more