On the Contractibility of Random Vietoris–Rips Complexes

We show that the Vietoris–Rips complex ℛ(n,r) built over n points sampled at random from a uniformly positive probability measure on a convex body K⊆ℝ^d is a.a.s. contractible when r≥ 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 ℛ(n,r) is a.a.s. homotopy equivalent to K when c_1(ln n/n)^1/d≤ r≤ 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.