  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 $n^{2/3}$.
