Randomized Optimal Transport on a Graph: framework and new distance measures

TL;DR

This paper extends the bag-of-paths framework to include constraints on start and end node distributions, linking it to optimal transport and electrical network analogies, and introduces new graph-based distance measures.

Contribution

It introduces a constrained bag-of-paths formalism that connects to optimal transport and electrical circuits, providing new distance measures and algorithms for path-based graph analysis.

Findings

The extended framework interpolates between shortest-path and resistance distances.

Algorithms for computing optimal free energy solutions are developed.

New node and group dissimilarity measures are derived from the framework.

Abstract

The recently developed bag-of-paths (BoP) framework consists in setting a Gibbs-Boltzmann distribution on all feasible paths of a graph. This probability distribution favors short paths over long ones, with a free parameter (the temperature $T$) controlling the entropic level of the distribution. This formalism enables the computation of new distances or dissimilarities, interpolating between the shortest-path and the resistance distance, which have been shown to perform well in clustering and classification tasks. In this work, the bag-of-paths formalism is extended by adding two independent equality constraints fixing starting and ending nodes distributions of paths (margins). When the temperature is low, this formalism is shown to be equivalent to a relaxation of the optimal transport problem on a network where paths carry a flow between two discrete distributions on nodes. The randomization is achieved by considering free energy minimization instead of traditional cost minimization. Algorithms computing the optimal free energy solution are developed for two types of paths: hitting (or absorbing) paths and non-hitting, regular, paths, and require the inversion of an $n \times n$ matrix with $n$ being the number of nodes. Interestingly, for regular paths on an undirected graph, the resulting optimal policy interpolates between the deterministic optimal transport policy ($T \rightarrow 0^{+}$) and the solution to the corresponding electrical circuit ($T \rightarrow \infty$). Two distance measures between nodes and a dissimilarity between groups of nodes, both integrating weights on nodes, are derived from this framework.