Efficient estimation of the modified Gromov-Hausdorff distance between unweighted graphs

TL;DR

This paper introduces a polynomial-time algorithm to estimate the modified Gromov-Hausdorff distance between unweighted graphs, enabling practical shape comparison and outlier detection in network analysis.

Contribution

The authors develop and implement a polynomial algorithm for estimating the modified Gromov-Hausdorff distance for unweighted graphs, making shape comparison computationally feasible.

Findings

Algorithm computes mGH distances accurately on scale-free graphs.

Effective outlier detection in real-world networks.

Performance demonstrated on synthetic and real data.

Abstract

Gromov-Hausdorff distances measure shape difference between the objects representable as compact metric spaces, e.g. point clouds, manifolds, or graphs. Computing any Gromov-Hausdorff distance is equivalent to solving an NP-Hard optimization problem, deeming the notion impractical for applications. In this paper we propose polynomial algorithm for estimating the so-called modified Gromov-Hausdorff (mGH) distance, whose topological equivalence with the standard Gromov-Hausdorff (GH) distance was established in M\'emoli F, 2012. We implement the algorithm for the case of compact metric spaces induced by unweighted graphs as part of Python library $\verb|scikit-tda|$, and demonstrate its performance on real-world and synthetic networks. The algorithm finds the mGH distances exactly on most graphs with the scale-free property. We use the computed mGH distances to successfully detect outliers in real-world social and computer networks.