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Bootstrap Percolation on the Binomial Random $k$-uniform Hypergraph

Mihyun Kang, Christoph Koch, Tamás Makai

Abstract

We investigate the behaviour of $r$-neighbourhood bootstrap percolation on the binomial $k$-uniform random hypergraph $H_k(n,p)$ for given integers $k\geq 2$ and $r\geq 2$. In $r$-neighbourhood bootstrap percolation, infection spreads through the hypergraph, starting from a set of initially infected vertices, and in each subsequent step of the process every vertex with at least $r$ infected neighbours becomes infected. For our analysis the set of initially infected vertices is chosen uniformly at random from all sets of given size. In the regime $n^{-1}\ll n^{k-2}p \ll n^{-1/r}$ we establish a threshold such that if the number of initially infected vertices remains below the threshold, then with high probability only a few additional vertices become infected, while if the number of initially infected vertices exceeds the threshold then with high probability almost every vertex becomes infected. In fact we show that the probability of failure decreases exponentially.

Bootstrap Percolation on the Binomial Random $k$-uniform Hypergraph

Abstract

We investigate the behaviour of -neighbourhood bootstrap percolation on the binomial -uniform random hypergraph for given integers and . In -neighbourhood bootstrap percolation, infection spreads through the hypergraph, starting from a set of initially infected vertices, and in each subsequent step of the process every vertex with at least infected neighbours becomes infected. For our analysis the set of initially infected vertices is chosen uniformly at random from all sets of given size. In the regime we establish a threshold such that if the number of initially infected vertices remains below the threshold, then with high probability only a few additional vertices become infected, while if the number of initially infected vertices exceeds the threshold then with high probability almost every vertex becomes infected. In fact we show that the probability of failure decreases exponentially.
Paper Structure (24 sections, 22 theorems, 169 equations, 2 figures, 1 algorithm)

This paper contains 24 sections, 22 theorems, 169 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction and main result
  2. Main result
  3. Related results
  4. Heuristics for the percolation threshold
  5. Proof outline
  6. Preliminaries
  7. Preliminaries for subcritical regime: the query-process
  8. An upper coupling
  9. The query-process
  10. Auxiliary results
  11. Coupling $r$-boostrap percolation and the query-process
  12. Subcritical regime
  13. Properties of the query-process
  14. Bounding trajectory
  15. Remaining below the trajectory
  16. ...and 9 more sections

Key Result

Theorem 1

Given an integer $r\geq 2$, consider $r$-bootstrap percolation on $G(n,p)$ when $n^{-1}\ll p\ll n^{-1/r}$. Assume that the initial infection set is chosen uniformly at random from the family of all sets of vertices of size $a=a(n)$. Then for any fixed $\varepsilon>0$, with probability at least $1-\e

Figures (2)

  • Figure 1: Collections of the query process. Filled circles represent infected vertices, empty circles represent uninfected vertices. Ovals represent exposed hyperedges, dashed ovals represent a collection of $k$-sets.
  • Figure 2: Potential parents. Empty circles represent uninfected vertices, while the filled circles represent infected vertices. Red circles were infected in step $t+1$, blue in step $t$ and black in the first $t-1$ steps.

Theorems & Definitions (61)

  • Theorem 1: MR3025687MR3817532
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • Theorem 5: Dwass Identity
  • Theorem 6: FKG inequality
  • Theorem 7: Azuma's inequality
  • Definition 8: Star collections; see Figure \ref{['fig:constructions']}(a)
  • Definition 9: Widely-overlapping collections; see Figure \ref{['fig:constructions']}(b)
  • Definition 10: Heavily-infected collections; see Figure \ref{['fig:constructions']}(c)
  • ...and 51 more