Maximally Persistent Cycles in Random Geometric Complexes

TL;DR

This paper investigates the behavior of the longest-lasting topological features in random geometric complexes generated from a Poisson point process, revealing their growth rate as the number of points increases.

Contribution

It provides the first probabilistic characterization of the persistence of maximal cycles in random geometric complexes, linking algebraic topology with geometric probability.

Findings

Maximal persistent cycles grow at a rate of Θ((log n / log log n)^{1/k})

Growth rate depends on dimension d and homology degree k

Results apply to both Čech and Vietoris--Rips filtrations

Abstract

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 \cech 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 (multiplicative) 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 \cech filtration.