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Large deviations for subcritical bootstrap percolation on the random graph

Omer Angel, Brett Kolesnik

TL;DR

This work analyzes rare events in subcritical bootstrap percolation on ${\mathcal{G}}_{n,p}$ by deriving an explicit large-deviation rate function $\xi(\alpha,\beta)$ for the final infected size when the initial susceptible set is of size $|S_0|\sim\alpha_r\vartheta$. Using a discretized Euler–Lagrange variational framework, it identifies the least-cost trajectory $\hat{y}_{\alpha,\beta}(x)$ that realizes such deviations, and shows the cost equals $\vartheta\xi(\alpha,\beta)$ with $\hat{y}_{\alpha,\beta}$ collapsing to the typical zero-cost path at $\beta=\varphi_\alpha$. The results yield monotonicity properties of $\xi$ and provide lower bounds for the minimal contagious set size in the subcritical regime, linking large-deviation principles to sharp thresholds and applications in contagion problems. These insights deepen understanding of rare but impactful spread phenomena in networks and offer tools for quantifying the costs of atypical propagation in random graphs.

Abstract

We study atypical behavior in bootstrap percolation on the Erdős-Rényi random graph. Initially a set $S$ is infected. Other vertices are infected once at least $r$ of their neighbors become infected. Janson et al. (2012) locates the critical size of $S$, above which it is likely that the infection will spread almost everywhere. Below this threshold, a central limit theorem is proved for the size of the eventually infected set. In this note, we calculate the rate function for the event that a small set $S$ eventually infects an unexpected number of vertices, and identify the least-cost trajectory realizing such a large deviation.

Large deviations for subcritical bootstrap percolation on the random graph

TL;DR

This work analyzes rare events in subcritical bootstrap percolation on by deriving an explicit large-deviation rate function for the final infected size when the initial susceptible set is of size . Using a discretized Euler–Lagrange variational framework, it identifies the least-cost trajectory that realizes such deviations, and shows the cost equals with collapsing to the typical zero-cost path at . The results yield monotonicity properties of and provide lower bounds for the minimal contagious set size in the subcritical regime, linking large-deviation principles to sharp thresholds and applications in contagion problems. These insights deepen understanding of rare but impactful spread phenomena in networks and offer tools for quantifying the costs of atypical propagation in random graphs.

Abstract

We study atypical behavior in bootstrap percolation on the Erdős-Rényi random graph. Initially a set is infected. Other vertices are infected once at least of their neighbors become infected. Janson et al. (2012) locates the critical size of , above which it is likely that the infection will spread almost everywhere. Below this threshold, a central limit theorem is proved for the size of the eventually infected set. In this note, we calculate the rate function for the event that a small set eventually infects an unexpected number of vertices, and identify the least-cost trajectory realizing such a large deviation.

Paper Structure

This paper contains 9 sections, 4 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Heuristics
  3. Motivation
  4. An application
  5. Proof of \ref{['T_LD']}
  6. Technical results
  7. Shape of $\xi$
  8. Lower bound for $m({\mathcal{G}}_{n,p},r)$
  9. Increments of $\pi$

Key Result

Theorem 1

Let $p$ be as in E_p and $\alpha\ge0$. Put $\alpha_r=(1-1/r)\alpha$. Suppose that a set $S_0$ (independent of ${\mathcal{G}}_{n,p}$) of size $|S_0|\sim\alpha_r\vartheta$ is initially susceptible. If $\alpha>1$, then with high probability $|I_*|\sim n$. If $\alpha<1$, then with high probability $|I_*

Figures (2)

  • Figure 1: For $r=2$ and $\alpha=2/3$, the rate function $-\xi(\alpha,\beta)$ is plotted as a function of $\beta$.
  • Figure 2: In both figures, $r=2$ and $\alpha=2/3$. The typical, zero-cost trajectory appears as a dotted line. Least-cost, deviating trajectories $\hat{y}_{\alpha,\beta}$ for $\beta=1/3,2/5$ appear at left and $\beta=1/2,2/3,5/6$ at right.

Theorems & Definitions (7)

  • Theorem 1: JLTV12 Theorem 3.1
  • Definition 2
  • Theorem 3
  • Corollary 4
  • Lemma 5
  • proof : Proof of \ref{['T_mGr']}
  • proof : Proof of \ref{['L_delpi']}