Probability Seminar
Friday, July 21, 2017, 3:30pm, 119 Physics
Gugan Thoppe (Technion - Israel Institute of Technology)
On Betti Numbers, Persistence Diagrams, and Minimal Spanning Acycles of Random Complexes
Abstract:
A simplicial complex generalizes a graph; it is a collection of subsets of a vertex set, which is closed under subset operation. Betti numbers and Persistence Diagram (PD) are topological descriptors of a simplicial complex; while Betti numbers count holes of different dimensions, PD tracks the birth and death instances of distinct topological features as the complex is sequentially built piece by piece. Separately, a Minimal Spanning Acycle (MSA) generalizes the notion of a minimal spanning tree to weighted simplicial complexes. This talk has four parts. In the first part, we shall briefly look at relevant notions from simplicial homology. In the second part, we will consider a time varying analogue of the Erdos-Renyi graph, which we refer to as the dynamic Erdos-Renyi graph. We shall prove that under suitable assumptions, scaled Betti numbers of this dynamic complex converge to the stationary Ornstein-Uhlenbeck process as the number of vertices tends to infinity. In the third part, we shall study the relationship between a MSA of a weighted simplicial complex and the associated PD. Using this, we will prove that the extremal weights in the MSA of a randomly weighted complex and extremal death times in the associated PD converge to a Poisson point process as the number of vertices go to infinity. In fact, we shall show that these results also hold if the random weights have small perturbations. In the fourth part, we shall look at Cech complexes on the excursion set of a random Gaussian field, whose covariance satisfies some local and decay rate conditions. We shall see the precise rate at which the window size and excursion level should grow to infinity so that, asymptotically, the different Betti numbers have a Poisson limit. We shall end with some future directions.

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