Comparing graphs 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.

翻译：比较图的最优传输距离近期受到显著关注，因为由最优传输诱导的距离不仅提供了图之间基于原理的度量标准，还通过传输方案对图之间的相关变化提供了可解释的描述。由于缺乏对称性给通常考虑的公式带来了挑战，图的最优传输距离大多针对无向图而开发。在此，我们提出了两种基于最优传输变体的距离度量来比较有向图：（i）推土机距离（Wasserstein）和（ii）Gromov-Wasserstein（GW）距离。我们评估了这两种距离，并讨论了它们在模拟图数据以及从单细胞RNA-seq数据推断的真实世界有向细胞间通信图中的相对性能。