Optimal Transport for Network Comparison: A Review with Machine Learning Applications
TL;DR
This paper examines the closed form of the Wasserstein distance in one dimension via node feature probability distributions, and shows how the transport plans of the Wasserstein and Gromov-Wasserstein distances visualize how mass is shifted to transform one network into another.
Abstract
Network comparison using optimal transport is a growing area of research in network science. Unlike standard graph metrics, optimal transport computes both network dissimilarity and a transport plan that explains how one graph morphs into another. In this paper, we review how optimal transport compares undirected, unweighted graphs using three primary distances: the Wasserstein, Gromov-Wasserstein, and Bures-Wasserstein distances. We examine the closed form of the Wasserstein distance in one dimension via node feature probability distributions, and show how the transport plans of the Wasserstein and Gromov-Wasserstein distances visualize how mass is shifted to transform one network into another. For the Bures-Wasserstein distance, we derive bounds in terms of the Laplacian spectra. Finally, we evaluate these distances using a synthetic network dataset for clustering and a real-world temporal network.
