Gromov Hyperbolicity, Geodesic Defect, and Apparent Pairs in Vietoris-Rips Filtrations

TL;DR

This paper extends the understanding of Vietoris-Rips complexes in geodesic and general metric spaces, linking hyperbolicity and geodesic defect to computational methods like Ripser for efficient topological data analysis.

Contribution

It generalizes Rips' contractibility result using geodesic defect, and connects collapses in Vietoris-Rips complexes to the apparent pairs gradient in Ripser.

Findings

Vietoris-Rips complexes of finite trees collapse to subforests.

Collapse induced by apparent pairs gradient explains Ripser's efficiency on tree data.

Generalization of contractibility results to broader metric spaces.

Abstract

Motivated by computational aspects of persistent homology for Vietoris-Rips filtrations, we generalize a result of Eliyahu Rips on the contractibility of Vietoris-Rips complexes of geodesic spaces for a suitable parameter depending on the hyperbolicity of the space. We consider the notion of geodesic defect to extend this result to general metric spaces in a way that is also compatible with the filtration. We further show that for finite tree metrics the Vietoris-Rips complexes collapse to their corresponding subforests. We relate our result to modern computational methods by showing that these collapses are induced by the apparent pairs gradient, which is used as an algorithmic optimization in Ripser, explaining its particularly strong performance on tree-like metric data.