Entropic Optimal Transport in Random Graphs

TL;DR

This paper demonstrates that entropic-regularized optimal transport distances between node groups in latent space can be consistently estimated from graph structures, with stability and concentration results applicable to various random graph models.

Contribution

It introduces a general stability result for entropic OT in random graphs and applies it to graphons and manifold-based graphs, along with new concentration inequalities.

Findings

Consistent estimation of entropic OT distances in latent space from graph data.

Stability of entropic OT with respect to cost matrix perturbations.

New concentration results for Universal Singular Value Thresholding and geodesic distance estimation.

Abstract

In graph analysis, a classic task consists in computing similarity measures between (groups of) nodes. In latent space random graphs, nodes are associated to unknown latent variables. One may then seek to compute distances directly in the latent space, using only the graph structure. In this paper, we show that it is possible to consistently estimate entropic-regularized Optimal Transport (OT) distances between groups of nodes in the latent space. We provide a general stability result for entropic OT with respect to perturbations of the cost matrix. We then apply it to several examples of random graphs, such as graphons or $\epsilon$-graphs on manifolds. Along the way, we prove new concentration results for the so-called Universal Singular Value Thresholding estimator, and for the estimation of geodesic distances on a manifold.