A sample-path large deviation principle for dynamic Erd\H{o}s-R\'enyi random graphs

TL;DR

This paper establishes a large deviation principle for the evolution of empirical graphons in a dynamic Erdős-Rényi graph model, revealing the most probable paths for atypical subgraph densities and transitions between graphons.

Contribution

It introduces a large deviation principle for the sample path of the empirical graphon in a dynamic ERRG, including analysis of bifurcations in variational solutions.

Findings

Large deviation principle with rate inom{n}{2} for graphon trajectories

Identification of most likely paths for atypical subgraph densities

Detection of bifurcations in variational problem solutions

Abstract

We consider a dynamic Erd\H{o}s-R\'enyi random graph (ERRG) on $n$ vertices in which each edge switches on at rate $\lambda$ and switches off at rate $\mu$, independently of other edges. The focus is on the analysis of the evolution of the associated empirical graphon in the limit as $n\to\infty$. Our main result is a large deviation principle (LDP) for the sample path of the empirical graphon observed until a fixed time horizon. The rate is $\binom{n}{2}$, the rate function is a specific action integral on the space of graphon trajectories. We apply the LDP to identify (i) the most likely path that starting from a constant graphon creates a graphon with an atypically large density of $d$-regular subgraphs, and (ii) the mostly likely path between two given graphons. It turns out that bifurcations may occur in the solutions of associated variational problems.