Gromov hyperbolicity, geodesic defect, and apparent pairs in Vietoris-Rips filtrations

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 introduce the notion of geodesic defect to extend this result to general metric spaces in a way that is also compatible with the Rips filtration. We further show that for finite tree metrics the Vietoris-Rips complexes collapse to their corresponding subtrees. 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.