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Phase transitions for contact processes on sparse random graphs via metastability and local limits

Benedikt Jahnel, Lukas Lüchtrath, Christian Mönch

TL;DR

We study fast versus slow extinction in the contact process on sequences of locally converging sparse graphs, linking finite-time dynamics to the local limit via the metastable density and the limit survival probability $\eta_\lambda(\mathsf{Q})$. Using local convergence in probability to an extremal limit $\mathsf{Q}$, we derive upper bounds on metastability and establish conditions under which $\lambda_+$ coincides with $\lambda_{1}(\mathscr{G})$; we also show that the natural exponential time scale governs the fast/slow separation in sparse graphs and construct examples where metastability fails on this scale. For configuration models, the fast/slow threshold equals the local-limit survival threshold on the limiting Bienaymé--Galton--Watson tree conditioned on non-extinction, i.e., $\lambda_+ = \lambda_{1}(\mathscr{T})$, with a corresponding almost-local criterion governing lower bounds. Overall, the work connects finite network dynamics to local-limit theory, providing a principled framework for phase-transition analysis in sparse random graphs and guiding the understanding of thresholds in configuration-model settings.

Abstract

We propose a new perspective on the asymptotic regimes of fast and slow extinction in the contact process on locally converging sequences of sparse finite graphs. We characterise the phase boundary by the existence of a metastable density, which makes the study of the phase transition particularly amenable to local-convergence techniques. We use this approach to derive general conditions for the coincidence of the critical threshold with the survival/extinction threshold in the local limit. We further argue that the correct time scale to separate fast extinction from slow extinction in sparse graphs is, in general, the exponential scale, by showing that fast extinction may occur on stretched exponential time scales in sparse scale-free spatial networks. Together with recent results by Nam, Nguyen and Sly (Trans. Am. Math. Soc. 375, 2022), our methods can be applied to deduce that the fast/slow threshold in sparse configuration models coincides with the survival/extinction threshold on the limiting Galton-Watson tree.

Phase transitions for contact processes on sparse random graphs via metastability and local limits

TL;DR

We study fast versus slow extinction in the contact process on sequences of locally converging sparse graphs, linking finite-time dynamics to the local limit via the metastable density and the limit survival probability . Using local convergence in probability to an extremal limit , we derive upper bounds on metastability and establish conditions under which coincides with ; we also show that the natural exponential time scale governs the fast/slow separation in sparse graphs and construct examples where metastability fails on this scale. For configuration models, the fast/slow threshold equals the local-limit survival threshold on the limiting Bienaymé--Galton--Watson tree conditioned on non-extinction, i.e., , with a corresponding almost-local criterion governing lower bounds. Overall, the work connects finite network dynamics to local-limit theory, providing a principled framework for phase-transition analysis in sparse random graphs and guiding the understanding of thresholds in configuration-model settings.

Abstract

We propose a new perspective on the asymptotic regimes of fast and slow extinction in the contact process on locally converging sequences of sparse finite graphs. We characterise the phase boundary by the existence of a metastable density, which makes the study of the phase transition particularly amenable to local-convergence techniques. We use this approach to derive general conditions for the coincidence of the critical threshold with the survival/extinction threshold in the local limit. We further argue that the correct time scale to separate fast extinction from slow extinction in sparse graphs is, in general, the exponential scale, by showing that fast extinction may occur on stretched exponential time scales in sparse scale-free spatial networks. Together with recent results by Nam, Nguyen and Sly (Trans. Am. Math. Soc. 375, 2022), our methods can be applied to deduce that the fast/slow threshold in sparse configuration models coincides with the survival/extinction threshold on the limiting Galton-Watson tree.

Paper Structure

This paper contains 12 sections, 11 theorems, 79 equations.

Table of Contents

  1. Introduction and main results
  2. The contact process.
  3. The fast/slow threshold of the contact process on random graphs.
  4. Overview of main results.
  5. Further discussion and related work.
  6. Overview of the proof section.
  7. Proofs
  8. Preparatory and auxiliary results
  9. Slow extinction of subcritical contact processes on scale-free spatial networks
  10. Upper bounds on the metastable density from local limit
  11. Lower bounds on the metastable density from local limits
  12. Fast extinction by absence of metastability

Key Result

Theorem 1.3

Assume that $(\mathscr{G}_n)_{n\in\mathbb{N}}$ is a sequence of connected locally finite graphs that converges locally in probability to some rooted locally finite random graph $(\mathscr{G},o)$ with extremal distribution $\mathsf Q$. Then, for any diverging sequence $(t(n))_{n\in\mathbb{N}}$ of tim In particular, if $\lambda<\lambda_1(\mathscr{G})\equiv\lambda_1(\mathsf Q)$, then

Theorems & Definitions (25)

  • Remark 1.1
  • Conjecture 1.2: nam_critical_2022
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • proof
  • Proposition 1.7
  • Corollary 1.8
  • proof
  • ...and 15 more