$\epsilon$-net Induced Lazy Witness Complexes on Graphs

arXiv2009.13071MaRDI QIDQ6350001

Debabrota Basu, Naheed Anjum Arafat, Stéphane Bressan

Publication date: 25 September 2020

Abstract: Computation of persistent homology of simplicial representations such as the Rips and the Cv{e}ch complexes do not efficiently scale to large point clouds. It is, therefore, meaningful to devise approximate representations and evaluate the trade-off between their efficiency and effectiveness. The lazy witness complex economically defines such a representation using only a few selected points, called landmarks. Topological data analysis traditionally considers a point cloud in a Euclidean space. In many situations, however, data is available in the form of a weighted graph. A graph along with the geodesic distance defines a metric space. This metric space of a graph is amenable to topological data analysis. We discuss the computation of persistent homologies on a weighted graph. We present a lazy witness complex approach leveraging the notion of $epsilon$-net that we adapt to weighted graphs and their geodesic distance to select landmarks. We show that the value of the $epsilon$ parameter of the $epsilon$-net provides control on the trade-off between choice and number of landmarks and the quality of the approximate simplicial representation. We present three algorithms for constructing an $epsilon$-net of a graph. We comparatively and empirically evaluate the efficiency and effectiveness of the choice of landmarks that they induce for the topological data analysis of different real-world graphs.

Has companion code repository: https://github.com/toggled/snap_ripser

This page was built for publication: $\epsilon$-net Induced Lazy Witness Complexes on Graphs