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On the evolution of topology in dynamic clique complexes

Published online by Cambridge University Press:  11 January 2017

Gugan C. Thoppe*
Affiliation:
Technion – Israel Institute of Technology
D. Yogeshwaran*
Affiliation:
Indian Statistical Institute
Robert J. Adler*
Affiliation:
Technion – Israel Institute of Technology
*
* Postal address: Faculty of Electrical Engineering, Technion, Haifa, 32000, Israel.
*** Postal address: Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore, 560059, India.
* Postal address: Faculty of Electrical Engineering, Technion, Haifa, 32000, Israel.

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=nα, where n is the number of vertices and α∈(-1/k, -1/(k + 1)), then the process of the normalised kth Betti number of these dynamic clique complexes converges weakly to the Ornstein–Uhlenbeck process as n→∞.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2017 

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