On the Contractibility of Random Vietoris–Rips Complexes

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 $$r\ge c({\ln n}/{n})^{1/d}$$ for a certain constant that depends on K and the probability measure used. This answers a question of Kahle (Discrete Comput. Geom. 45(3), 553–573 (2011)). 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 $$c_1(\ln n/{n})^{1/d} \le r\le c_2$$ for constants $$c_1=c_1(K)$$ , $$c_2=c_2(K)$$ . Our proofs expose a connection with the game of cops and robbers.