The spread of fire on a random multigraph

TL;DR

The paper analyzes how fire spreads on a random multigraph with $n$ degree-3 vertices and $\alpha(n)$ degree-4 vertices, where edge lengths are i.i.d. exponential. Using a Markov-chain framework, a two-phase, differential-equations approach, and a coupling to a reflected random walk, it derives scaling limits for the number of fires $F^n$ and clashes $C^n$ as $n\to\infty$, with distinct regimes depending on the growth of $\alpha(n)$ relative to $\sqrt{n}$. If $\alpha(n)=o(\sqrt{n})$, the pair $(F^n,C^n)$, rescaled by appropriate factors, converges to constants or Brownian-functionals defined by a reflected diffusion; if $\alpha(n)\gg\sqrt{n}$, one has $F^n/\alpha(n)\to0$ and $C^n/\alpha(n)\to1/4$, with the intermediate, critical case $\alpha(n)\sim a\sqrt{n}$ yielding a diffusion limit $(X^a,L^a)$. The results contribute to understanding the Karp–Sipser matching process on sparse random graphs, offering progress toward Aronson–Frieze–Pittel's conjecture on unmatched vertices by connecting the fire-spread mechanism to the structure of complex components in Phase II. Overall, the work demonstrates how stochastic-analysis tools—fluid limits and diffusion limits via a reflected SDE—capture the random, combinatorial dynamics of burning and matching on random graphs.

Abstract

We study a model for the destruction of a random network by fire. Suppose that we are given a multigraph of minimum degree at least 2 having real-valued edge-lengths. We pick a uniform point from along the length and set it alight; the edges of the multigraph burn at speed 1. If the fire reaches a vertex of degree 2, the fire gets directly passed on to the neighbouring edge; a vertex of degree at least 3, however, passes the fire either to all of its neighbours or none, each with probability $1/2$. If the fire goes out before the whole network is burnt, we again set fire to a uniform point. We are interested in the number of fires which must be set in order to burn the whole network, and the number of points which are burnt from two different directions. We analyse these quantities for a random multigraph having $n$ vertices of degree 3 and $α(n)$ vertices of degree 4, where $α(n)/n \to 0$ as $n \to \infty$, with i.i.d. standard exponential edge-lengths. Depending on whether $α(n) \gg \sqrt{n}$ or $α(n)=O(\sqrt{n})$, we prove that as $n \to \infty$ these quantities converge jointly in distribution when suitably rescaled to either a pair of constants or to (complicated) functionals of Brownian motion. We use our analysis of this model to make progress towards a conjecture of Aronson, Frieze and Pittel concerning the number of vertices which remain unmatched when we use the Karp-Sipser algorithm to find a matching on the Erdős-Rényi random graph.

Theorems & Definitions (37)

• Theorem 1
• Theorem 2
• Proposition 3
• Theorem 4
• Lemma 5
• proof
• Proposition 6
• proof
• Lemma 7
• proof