Exceptional times of the critical dynamical Erd.s.Rényi graph

In this paper, we introduce a network model which evolves in time, and study its largest connected component. We consider a process of graphs (Gt:t.[0,1]), where initially we start with a critical Erd.s.Rényi graph ER(n,1/n), and then evolve forward 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 n2/3log1/3n with high probability. This is in contrast to the largest component in the static critical Erd.s.Rényi graph, which is of order n2/3.