On the evolution of topology in dynamic clique complexes
Gugan Thoppe, Dhandapani Yogeshwaran, Robert J. Adler · Advances in Applied Probability · 2016
Abstract We consider a time varying analogue of the Erdős–Rényi 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 formp=nα, wherenis the number of vertices and α∈(-1/k, -1/(k+ 1)), then the process of the normalisedkth Betti number of these dynamic clique complexes converges weakly to the Ornstein–Uhlenbeck process asn→∞.