On the evolution of topology in dynamic clique complexes

Abstract

We consider a time varying analogue of the Erd{\H o}s-R{\' e}nyi graph and study the topological variations of its associated clique complex. The dynamics of the graph are stationary and are determined by the edges, which evolve independently as continuous time Markov chains. Our main result is that when the edge inclusion probability is of the form $p = n^\alpha$, where $n$ is the number of vertices and $\alpha \in (-1/k, -1/(k + 1)),$ then the process of the normalized $k-$th Betti number of these dynamic clique complexes converges weakly to the Ornstein-Uhlenbeck process as $n \to \infty.$