Homological Properties of Random and Geometric Clique Complexes

Abstract: 

Clique topology provides a framework for studying the geometric structure of data through the topology of simplicial complexes. It can, for example, be used to detect geometric structure in neural activity in a way that is invariant under nonlinear monotone transformations. This seminar will explore the mathematical foundations of clique topology, focusing on the behaviour of Betti numbers and Betti curves in random complexes.

We will consider two important classes of random simplicial complexes: random clique complexes arising from the Erdős-Rényi model of random graphs, and random Vietoris-Rips complexes. Following the results of Matthew Kahle, we will examine the phase transitions between vanishing and non-vanishing homology in these complexes. We will also consider the critical regime in which homology is non-vanishing. In this regime, we will compare the asymptotic behaviour of Betti numbers in the two settings: for random Vietoris-Rips complexes, they grow linearly with the number of vertices, while for random clique complexes, the k-th Betti number is asymptotically equivalent to the number of k-faces.