On the evolution of topology in dynamic clique complexes. (11th January 2017)

• Record Type: Journal Article
• Authors: Thoppe, Gugan C.; Yogeshwaran, D.; Adler, Robert J.
• Is Part Of: Advances in applied probability. Volume 48:Number 4(2016)
• Page Start: 989
• Page End: 1014
• Publication Date: 2017-01-11
• Subjects: Dynamic Erdős–Rényi graph, -- Betti numbers, -- Ornstein–Uhlenbeck; Primary 05C80, -- Secondary 60C05, -- 55U10, -- 60B10; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2016.62
• Languages: English
• ISSNs: 0001-8678

Abstract

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 form p = n α, where n is the number of vertices and α∈(-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 →∞.