A sample-path large deviation principle for dynamic Erdős–Rényi random graphs

Abstract

We consider a dynamic Erdős–Rényi random graph on n vertices in which each edge switches on at rate λ and switches off at rate μ, independently of other edges. The focus is on the analysis of the evolution of the associated empirical graphon in the limit as $\mathit{n}\to \infty$. 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 $\left(\begin{array}{c}\mathit{n}\\ 2\end{array}\right)$, 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.