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On the evolution of topology in dynamic clique complexes

Part of: Applied homological algebra and category theory Probability theory on algebraic and topological structures Graph theory Combinatorial probability

Published online by Cambridge University Press:  11 January 2017

Gugan C. Thoppe ,
D. Yogeshwaran  and
Robert J. Adler
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.
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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→∞.

Keywords

Dynamic Erdős–Rényi graphBetti numbersOrnstein–Uhlenbeck

MSC classification

Primary: 05C80: Random graphs
Secondary: 60C05: Combinatorial probability 55U10: Simplicial sets and complexes 60B10: Convergence of probability measures
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 4 , December 2016 , pp. 989 - 1014
Copyright
Copyright © Applied Probability Trust 2017 

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