Vietoris–Rips complex

); if a finite set S has the property that the distance between every pair of points in S is at most δ, then we include S as a simplex in the complex.

[5] Chambers, Erickson & Worah (2008) describe efficient algorithms for determining whether a given cycle is contractible in the Rips complex of any finite point set in the Euclidean plane.

As with unit disk graphs, the Vietoris–Rips complex has been applied in computer science to model the topology of ad hoc wireless communication networks.

One advantage of the Vietoris–Rips complex in this application is that it can be determined only from the distances between the communication nodes, without having to infer their exact physical locations.

[8] The collection of all Vietoris–Rips complexes is a commonly applied construction in persistent homology and topological data analysis, and is known as the Rips filtration.

A Vietoris–Rips complex of a set of 23 points in the Euclidean plane . This complex has sets of up to four points: the points themselves (shown as red circles), pairs of points (black edges), triples of points (pale blue triangles), and quadruples of points (dark blue tetrahedrons).