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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