On the contractibility of random Vietoris-Rips complexes

TL;DR

This paper proves that random Vietoris-Rips complexes become contractible or homotopy equivalent to the underlying space under certain scale conditions, answering a key open question in topological data analysis.

Contribution

It establishes precise conditions under which Vietoris-Rips complexes are contractible or homotopy equivalent to the underlying convex body or manifold, extending previous results.

Findings

Vietoris-Rips complex is a.a.s. contractible for r ≥ c (ln n / n)^{1/d}

Complex is a.a.s. homotopy equivalent to the underlying manifold for specific r ranges

Connections with the game of cops and robbers are revealed in the proofs.

Abstract

We show that the Vietoris-Rips complex $\mathcal R(n,r)$ built over $n$ points sampled at random from a uniformly positive probability measure on a convex body $K\subseteq \mathbb R^d$ is a.a.s. contractible when $r \geq c \left(\frac{\ln n}{n}\right)^{1/d}$ for a certain constant that depends on $K$ and the probability measure used. This answers a question of Kahle [Discrete Comput. Geom. 45 (2011), 553-573]. We also extend the proof to show that if $K$ is a compact, smooth $d$-manifold with boundary - but not necessarily convex - then $\mathcal R(n,r)$ is a.a.s. homotopy equivalent to $K$ when $c_1 \left(\frac{\ln n}{n}\right)^{1/d} \leq r \leq c_2$ for constants $c_1=c_1(K), c_2=c_2(K)$. Our proofs expose a connection with the game of cops and robbers.