Graph Optimal Transport with Transition Couplings of Random Walks

Abstract

We present a novel approach to optimal transport between graphs from the perspective of stationary Markov chains. A weighted graph may be associated with a stationary Markov chain by means of a random walk on the vertex set with transition distributions depending on the edge weights of the graph. After drawing this connection, we describe how optimal transport techniques for stationary Markov chains may be used in order to perform comparison and alignment of the graphs under study. In particular, we propose the graph optimal transition coupling problem, referred to as GraphOTC, in which the Markov chains associated to two given graphs are optimally synchronized to minimize an expected cost. The joint synchronized chain yields an alignment of the vertices and edges in the two graphs, and the expected cost of the synchronized chain acts as a measure of distance or dissimilarity between the two graphs. We demonstrate that GraphOTC performs equal to or better than existing state-of-the-art techniques in graph optimal transport for several tasks and datasets. Finally, we also describe a generalization of the GraphOTC problem, called the FusedOTC problem, from which we recover the GraphOTC and OT costs as special cases.Preprint. Under review.

fGOT: Graph Distances Based on Filters and Optimal Transport

“…The approach consists of producing a discrete permutation from a continuous doubly-stochastic matrix obtained with the Sinkhorn operator (Sinkhorn 1964). More recently, optimal transport between stationary Markov chains has been used to find a coupling between graphs (O'Connor et al 2021).…”

Estimation of Stationary Optimal Transport Plans

“…In another direction, [53] studied computational aspects of a constrained form of the optimal joining problem for Markov chains. This constrained optimal joining problem was applied to the comparison and alignment of graphs in [54].…”