Vietoris–Rips complex

In topology, the Vietoris–Rips complex, also called the Vietoris complex or Rips complex, is an abstract simplicial complex that can be defined from any metric space M and distance δ by forming a simplex for every finite set of points that has diameter at most δ. That is, it is a family of finite subsets of M, in which we think of a subset of k points as forming a (k − 1)-dimensional simplex (an edge for two points, a triangle for three points, a tetrahedron for four points, etc.); 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.

The Vietoris–Rips complex was originally called the Vietoris complex, for Leopold Vietoris, who introduced it as a means of extending homology theory from simplicial complexes to metric spaces. After Eliyahu Rips applied the same complex to the study of hyperbolic groups, its use was popularized by Gromov (1987), who called it the Rips complex. The name "Vietoris–Rips complex" is due to Hausmann (1995).

The Vietoris–Rips complex is closely related to the Čech complex (or nerve) of a set of balls, which has a simplex for every finite subset of balls with nonempty intersection: in a geodesically convex space Y, the Vietoris–Rips complex of any subspace X ⊂ Y for distance δ has the same points and edges as the Čech complex of the set of balls of radius δ/2 in Y that are centered at the points of X. However, unlike the Čech complex, the Vietoris–Rips complex of X depends only on the intrinsic geometry of X, and not on any embedding of X into some larger space.

As an example, consider the uniform metric space M₃ consisting of three points, each at unit distance from each other. The Vietoris–Rips complex of M₃, for δ = 1, includes a simplex for every subset of points in M₃, including a triangle for M₃ itself. If we embed M₃ as an equilateral triangle in the Euclidean plane, then the Čech complex of the radius-1/2 balls centered at the points of M₃ would contain all other simplexes of the Vietoris–Rips complex but would not contain this triangle, as there is no point of the plane contained in all three balls. However, if M₃ is instead embedded into a metric space that contains a fourth point at distance 1/2 from each of the three points of M₃, the Čech complex of the radius-1/2 balls in this space would contain the triangle. Thus, the Čech complex of fixed-radius balls centered at M₃ differs depending on which larger space M₃ might be embedded into, while the Vietoris–Rips complex remains unchanged.

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