A new relationship between Erd\H{o}s-R\'{e}nyi graphs, epidemic models and Brownian motion with parabolic drift

TL;DR

This paper establishes a novel connection between Erdős-Rényi graphs, epidemic models, and Brownian motion with parabolic drift, revealing new scaling limits of epidemic-related statistics in random graphs.

Contribution

It introduces a new relationship linking epidemic models, Erdős-Rényi graphs, and stochastic processes, providing insights into their asymptotic behavior in critical regimes.

Findings

Weak convergence to Brownian motion with parabolic drift in the critical window

Deterministic scaling limit outside the critical window

Unified framework for epidemic statistics on random graphs

Abstract

In the Reed-Frost model, an example of an SIR epidemic model, one can examine a statistic that counts the number of concurrently infected individuals. This statistic can be reformulated as a statistic on the \ER random graph $G(n,p)$. Within the critical window of Aldous and Martin-L\"{o}f, i.e. when $p = p(n) = n^{-1}+\lambda n^{-4/3}$, the cumulative sum of this statistic converges weakly to the integral of a Brownian motion with parabolic drift. This same statistic exhibits a deterministic scaling limit when $p = (1+\lambda \varepsilon_n)/n$ whenever $\varepsilon_n\to 0$ and $n^{1/3}\varepsilon_n\to\infty$.