Cech versus Rips Triangle Hole

In topological data analysis, we often build simplicial complexes from a set of points in a metric space to study their shape. Two common constructions are the Čech complex and the Vietoris-Rips complex. Both depend on a radius parameter r.

worked-example

Consider three points forming an equilateral triangle of side length d = 1. We place a closed ball of radius r at each point. Let's analyze the complexes when r = 11/20 = 0.55.

Edges: An edge exists between two points if the distance between them is at most 2r. Since d = 1 and 2r = 1.1, the distance is less than 2r, so all three edges exist.
Čech face: A face exists in the Čech complex if all three closed balls share a common point of intersection. The minimum radius for this to happen is the circumradius R of the triangle, which is d / sqrt(3) ≈ 0.577. Since r = 0.55 < R, the three balls do not have a common intersection. The Čech complex has 0 faces and thus contains a "hole" (Betti 1 = 1).
Rips face: The Vietoris-Rips complex adds a face whenever all its edges are present (it is a clique complex). Since all 3 edges are present, the Rips face is added. The Rips complex has 1 face and no hole (Betti 1 = 0).

experiment

Use the Cech versus Rips Triangle Hole module controls to explore how the complexes evolve:

Set the Triangle side to 1.
Set the Ball radius ratio to 1/2. Observe that edges appear in both complexes. The Rips complex immediately fills in the face because it's a clique complex, so its 1-dimensional homology (Betti 1) is 0.
Set the Ball radius ratio to 11/20. The Čech complex still has a hole (Betti 1 = 1) because the balls do not share a common intersection point.
Set the Ball radius ratio to 1/sqrt(3). Observe that the Čech complex finally fills the face, as the balls now share a common intersection point at the circumcenter. Both complexes now have Betti 1 = 0.

limitations

This module uses a highly symmetric configuration of three points. In real-world data, points are rarely arranged in exact equilateral triangles. The discrepancies between Čech and Rips complexes can occur in much more complex and higher-dimensional ways. Additionally, while the Rips complex is computationally easier to construct (requiring only pairwise distances), the Čech complex is guaranteed to be homotopy equivalent to the union of the balls (by the Nerve Theorem), which the Rips complex is not.

references

The properties of Cech and Vietoris-Rips complexes, including the differences in their threshold radii and applications in topological data analysis, are fundamental concepts covered in computational topology literature. They illustrate the trade-off between geometric fidelity and computational efficiency when analyzing the shape of data.

Computational Topology, Lecture sec-III-2

Cech versus Rips Triangle Hole