Optimal transport distances for directed, weighted graphs: a case study with cell-cell communication networks

TL;DR

This paper introduces two optimal transport-based distance measures for directed, weighted graphs, specifically applied to cell-cell communication networks, addressing symmetry challenges and evaluating their effectiveness on real and simulated data.

Contribution

It proposes novel optimal transport distances tailored for directed graphs, extending existing methods primarily designed for undirected graphs.

Findings

Wasserstein and Gromov-Wasserstein distances effectively compare directed graphs.

The methods reveal meaningful differences in cell-cell communication networks.

Performance varies between simulated and real-world data.

Abstract

Comparing graphs by means of optimal transport has recently gained significant attention, as the distances induced by optimal transport provide both a principled metric between graphs as well as an interpretable description of the associated changes between graphs in terms of a transport plan. As the lack of symmetry introduces challenges in the typically considered formulations, optimal transport distances for graphs have mostly been developed for undirected graphs. Here, we propose two distance measures to compare directed graphs based on variants of optimal transport: (i) an earth movers distance (Wasserstein) and (ii) a Gromov-Wasserstein (GW) distance. We evaluate these two distances and discuss their relative performance for both simulated graph data and real-world directed cell-cell communication graphs, inferred from single-cell RNA-seq data.