A Dynamic Erd\H{o}s-R\'enyi Graph Model

TL;DR

This paper introduces a dynamic Erdős-Rényi graph model where edges evolve via Markov processes, providing explicit formulas for mixing times and edge count reaching times, with analysis of their asymptotic behavior.

Contribution

It presents a novel dynamic graph model with explicit analysis of stationarity and edge accumulation times, advancing understanding of temporal network evolution.

Findings

Explicit expression for the time to reach stationarity.

Proof that this is the fastest strong stationary time.

Asymptotic analysis of the time to reach a certain number of edges.

Abstract

In this article we introduce a dynamic Erd\H{o}s-R\'enyi graph model, in which, independently for each vertex pair, edges appear and disappear according to a Markov on-off process. In studying the dynamic graph we present two main results. The first being on how long it takes for the graph to reach stationarity. We give an explicit expression for this time, as well as proving that this is the fastest time to reach stationarity among all strong stationary times. The second result concerns the time it takes for the dynamic graph to reach a certain number of edges. We give an explicit expression for the expected value of such a time, as well as study its asymptotic behavior. This time is related to the first time the dynamic Erd\H{o}s-R\'enyi graph contains a cluster exceeding a certain size.