Exceptional times of the critical dynamical Erd\H{o}s-R\'enyi graph

Abstract

In this paper we introduce a network model which evolves in time, and study its largest connected component. We consider a process of graphs $(G_t:t\in [0,1])$, where initially we start with a critical Erd\H{o}s-R\'enyi graph ER(n, 1/n), and then evolve forwards in time by resampling each edge independently at rate 1. We show that the size of the largest connected component that appears during the time interval $[0, 1]$ is of order $n^{2/3} log^{1/3} n$ with high probability. This is in contrast to the largest component in the static critical Erd\H{o}s-R\'enyi graph, which is of order $n^{2/3}$.