On the evolution of topology in dynamic clique complexes

TL;DR

This paper studies how the topology of clique complexes derived from a dynamic Erdős-Rényi graph evolves over time, showing convergence of Betti numbers to an Ornstein-Uhlenbeck process under certain conditions.

Contribution

It introduces a dynamic model for clique complexes based on evolving Erdős-Rényi graphs and proves the weak convergence of Betti numbers to a continuous stochastic process.

Findings

Betti numbers of the dynamic clique complex converge to an Ornstein-Uhlenbeck process.

The convergence occurs when edge probability scales as a power of the number of vertices.

The model captures topological phase transitions in evolving random complexes.

Abstract

We consider a time varying analogue of the Erd{\H o}s-R{\' e}nyi 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 normalized $k-$th Betti number of these dynamic clique complexes converges weakly to the Ornstein-Uhlenbeck process as $n \to \infty.$