Maximally Persistent Cycles in Random Geometric Complexes

Omer Bobrowski, Matthew Kahle, Primoz Skraba

Published 2015-09-14Version 1

We initiate the study of persistent homology of random geometric simplicial complexes. Our main interest is in maximally persistent cycles of degree $k$ in persistent homology, for a either the \v{C}ech or the Vietoris-Rips filtration built on a uniform Poisson process of intensity $n$ in the unit cube $[0,1]^d$. This is a natural way of measuring the largest "$k$-dimensional hole" in a random point set. This problem is in the intersection of geometric probability and algebraic topology, and is naturally motivated by a probabilistic view of topological inference. We show that for all $d \ge 2$ and $1 \le k \le d-1$ the maximally persistent cycle has persistence of order $$ \Theta \left( \left( \frac{\log n}{\log \log n} \right)^{1/k} \right),$$ with high probability, characterizing its rate of growth as $n \to \infty$. The implied constants depend on $k$, $d$, and on whether we consider the Vietoris-Rips or \v{C}ech filtration.