Dynamic Erd\H{o}s-R\'enyi graphs

TL;DR

This paper introduces two dynamic models of Erdős-Rényi graphs with evolving edge probabilities, providing explicit results on their statistical properties, including moments, limit theorems, and large deviations.

Contribution

It presents novel dynamic variants of Erdős-Rényi graphs governed by external regimes or periodic resampling, with detailed probabilistic analysis.

Findings

Explicit formulas for moments of the number of edges.

Functional central limit theorems established.

Large deviations asymptotics derived.

Abstract

We propose two classes of dynamic versions of the classical Erd\H{o}s-R\'enyi graph: one in which the transition rates are governed by an external regime process, and one in which the transition rates are periodically resampled. For both models we consider the evolution of the number of edges present, with explicit results for the corresponding moments, functional central limit theorems and large deviations asymptotics.