On the evolution of topology in dynamic clique complexes. (English) Zbl 1356.05136

Thoppe, Gugan C.; Yogeshwaran, D.; Adler, Robert J.

Adv. Appl. Probab. 48, No. 4, 989-1014 (2016).

Summary: 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 form \(p=n^\alpha\), where \(n\) is the number of vertices and \(\alpha \in (-1/k, -1/(k+1))\), then the process of the normalised \(k\)th Betti number of these dynamic clique complexes converges weakly to the Ornstein-Uhlenbeck process as \(n\).

MSC:

05C80	Random graphs (graph-theoretic aspects)
60C05	Combinatorial probability
55U10	Simplicial sets and complexes in algebraic topology
60B10	Convergence of probability measures
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords:

dynamic Erdős-Rényi graph; Betti numbers; Ornstein-Uhlenbeck