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Comparing graphs by means of optimal transport has recently gainedsignificant attention, as the distances induced by optimal transport provideboth a principled metric between graphs as well as an interpretable descriptionof the associated changes between graphs in terms of a transport plan. As thelack of symmetry introduces challenges in the typically consideredformulations, optimal transport distances for graphs have mostly been developedfor undirected graphs. Here, we propose two distance measures to comparedirected graphs based on variants of optimal transport: (i) an earth moversdistance (Wasserstein) and (ii) a Gromov-Wasserstein (GW) distance. We evaluatethese two distances and discuss their relative performance for both simulatedgraph data and real-world directed cell-cell communication graphs, inferredfrom single-cell RNA-seq data.

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