# Vietoris-Rips Complexes

#### Main Theorem

One of the most common "thickenings" of a point cloud that is used in toological data analysis is the following, because of its computational efficiency:

Definition. For a finite metric space $$X$$, and a scale parameter $$r > 0$$, the Vietoris-Rips Complex (or simply, VR Complex), denoted by $$VR(X;r)$$ is the complex that contains a finite simplex $$\sigma \subseteq X$$ whenever diam$$(\sigma) < r.$$

Here we make three important remarks about VR Complexes:

Remark 1. For every scale parameter $$r \ge 0$$, the complex $$VR(X;r)$$ is simplicial. That is, it satisifes the properties: (1) If $$\sigma$$ is in $$VR(X;r)$$ and if $$\tau$$ is in $$\sigma$$, then $$\tau$$ is in $$VR(X;r)$$, and (2) If $$\sigma$$ and $$\tau$$ are in V$$R(X;r)$$, then $$\sigma \cap \tau$$ is in $$VR(X;r)$$.

Remark 2. For any scale parameters $$r' \ge r > 0$$, the natural inclusion map $$VR(X;r) \to VR(X;r')$$ is a homotopy equivalence. That is, any inceasing sequence of scale parameters gives rise to an increasing sequence of simplicial complexes, called a VR simplicial filtration.

Remark 3. The complex $$VR(X;r)$$ depends only on pairwise distances in $$X$$. That is, $$VR(X;r)$$ is equal to the clique complex of its 0-dimensional and 1-dimensional simplices.

Remarks 1 and 2 are necessary in a technical sense because they prove that one can examine the persistent homology the point cloud. Remark 3 is important because it allows us to use general tools about clique complexes to provide insight into VR complexes. Moreover, Remark 3 gives us an optimization for storing the combinatorial object $$VR(X;r)$$ in a computer program.

The importance of the VR simplicial filtration is solidified by an important theorem, proving that the resulting persistent homology barcodes from a VR simplicial filtration are "stable".

Theorem. For $$M$$ a compact metric space and $$X \subseteq M$$ a finite subspace, if $$X$$ converges to $$M$$ in the Gromov-Hausdorff distance, then $$PH(X)$$ converges to $$PH(M)$$ in the bottleneck distance.

One often likes to imagine that a point cloud $$X$$ was created through a (noisy) sampling from a manifold $$M$$. However, there are very few infinite, continuous spaces $$M$$ for which the barcodes $$PH(M)$$ are known. Towards this end, we were able to compute $$PH(M)$$ for $$M = P_n$$ the boundary of any regular polygon, equipped with the Euclidean metric of the plane.