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

TL;DR

This paper introduces an $\\epsilon$-net based lazy witness complex method for efficient topological data analysis of weighted graphs, balancing computational efficiency and accuracy in persistent homology computation.

Contribution

It adapts the $\\epsilon$-net concept to weighted graphs and proposes three algorithms for landmark selection, enhancing scalable topological analysis of graph data.

Findings

The $\\epsilon$ parameter controls landmark quantity and representation quality.

The proposed algorithms effectively select landmarks for various real-world graphs.

Empirical results show improved efficiency and comparable accuracy in persistent homology computation.

Abstract

Computation of persistent homology of simplicial representations such as the Rips and the C\v{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.