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.