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Thresholds for contagious sets in random graphs

Omer Angel, Brett Kolesnik

TL;DR

This work determines sharp thresholds for the susceptibility of ${\cal G}_{n,p}$ under $r$-bootstrap percolation, pinpointing a critical constant $\alpha_r$ in the regime $p=(\alpha/(n\log^{r-1}n))^{1/r}$ that separates whp non-percolation from whp percolation via a minimal contagious set of size $r$. The authors connect these thresholds to a time-dependent non-homogeneous branching process and derive explicit survival asymptotics, $\Pr(\text{survival})=\exp(-\tfrac{(r-1)^2}{r}k_r(1+o(1)))$, with $k_r=((r-1)!/\varepsilon)^{1/(r-1)}$ and $\varepsilon=np^r$. They further obtain an upper bound for $p_c(n,K_4)$ improving prior work and conjecture its asymptotic sharpness. The analysis introduces triangle-free constrained percolations ($\hat{r}$-percolations) to manage dependencies, employs the second moment method, and leverages Mantel-type results to establish approximate independence. Finally, the study extends to general $H$-bootstrap percolation with precise thresholds $p_c(n,H,r)$ and specializes to $K_4$ as a key case, yielding both a broad framework and concrete, sharp thresholds with potential for refinement.

Abstract

For fixed $r\geq 2$, we consider bootstrap percolation with threshold $r$ on the Erdős-Rényi graph ${\cal G}_{n,p}$. We identify a threshold for $p$ above which there is with high probability a set of size $r$ which can infect the entire graph. This improves a result of Feige, Krivelevich and Reichman, which gives bounds for this threshold, up to multiplicative constants. As an application of our results, we also obtain an upper bound for the threshold for $K_4$-bootstrap percolation on ${\cal G}_{n,p}$, as studied by Balogh, Bollobás and Morris. We conjecture that our bound is asymptotically sharp. These thresholds are closely related to the survival probabilities of certain time-varying branching processes, and we derive asymptotic formulae for these survival probabilities which are of interest in their own right.

Thresholds for contagious sets in random graphs

TL;DR

This work determines sharp thresholds for the susceptibility of under -bootstrap percolation, pinpointing a critical constant in the regime that separates whp non-percolation from whp percolation via a minimal contagious set of size . The authors connect these thresholds to a time-dependent non-homogeneous branching process and derive explicit survival asymptotics, , with and . They further obtain an upper bound for improving prior work and conjecture its asymptotic sharpness. The analysis introduces triangle-free constrained percolations (-percolations) to manage dependencies, employs the second moment method, and leverages Mantel-type results to establish approximate independence. Finally, the study extends to general -bootstrap percolation with precise thresholds and specializes to as a key case, yielding both a broad framework and concrete, sharp thresholds with potential for refinement.

Abstract

For fixed , we consider bootstrap percolation with threshold on the Erdős-Rényi graph . We identify a threshold for above which there is with high probability a set of size which can infect the entire graph. This improves a result of Feige, Krivelevich and Reichman, which gives bounds for this threshold, up to multiplicative constants. As an application of our results, we also obtain an upper bound for the threshold for -bootstrap percolation on , as studied by Balogh, Bollobás and Morris. We conjecture that our bound is asymptotically sharp. These thresholds are closely related to the survival probabilities of certain time-varying branching processes, and we derive asymptotic formulae for these survival probabilities which are of interest in their own right.

Paper Structure

This paper contains 25 sections, 25 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Bootstrap percolation
  3. Graph bootstrap percolation and seeds
  4. A non-homogeneous branching process
  5. Outline of the proof
  6. Lower bound for $p_c(n,r)$
  7. Small susceptible graphs
  8. Upper bounds for susceptible graphs
  9. Susceptible subgraphs of ${\cal G}_{n,p}$
  10. Sub-critical bounds
  11. Upper bound for $p_c(n,r)$
  12. Triangle-free susceptible graphs
  13. $\hat{r}$-bootstrap percolation on ${\cal G}_{n,p}$
  14. Super-critical bounds
  15. $\hat{r}$-percolations are almost independent
  16. ...and 10 more sections

Key Result

Theorem 1.1

Fix $r\ge2$ and $\alpha>0$. Let and denote If $\alpha>\alpha_r$, then with high probability ${\cal G}_{n,p}$ is susceptible. If $\alpha<\alpha_r$, then there exists $\beta=\beta(\alpha,r)$ so that for $G={\cal G}_{n,p}$, with high probability for every $I$ of size $r$ we have $|\langle I,G \rangle_r| \leq \beta\log n$. In particular, as $n\t

Theorems & Definitions (82)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.3
  • proof
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 72 more