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

Yet, extending OT-based distances to directed graphs is not simple, as a Comparing graphs by means of optimal transport has recently symmetric distance metric, typically derived from the distances between gained significant attention, as the distances induced by optimal nodes in the graph, is required within the cost function(s) typically transport provide both a principled metric between graphs as well employed within OT. To address this problem, here we consider as an interpretable description of the associated changes between two node-to-node distances, which have been developed for directed graphs in terms of a transport plan. As the lack of symmetry introduces graphs, namely, the Generalized Effective Resistance (GRD) [6] and challenges in the typically considered formulations, optimal Markov chain hitting time (HTD) [7]. Employing these distance transport distances for graphs have mostly been developed for undirected measures enables us to compute OT-based graph distances even for graphs. Here, we propose two distance measures to compare directed graphs. Specifically, we explore the use of these metrics for directed graphs based on variants of optimal transport: (i) an earth both Wasserstein (Earth Mover) and Gromov-Wasserstein based OT movers distance (Wasserstein) and (ii) a Gromov-Wasserstein (GW) distances for graphs [5].