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

James S. Nagai, Ivan G. Costa, Michael T. Schaub

TL;DR

Two distance measures to compare directed graphs based on variants of optimal transport based on variants of optimal transport(OT) are proposed: an earth movers distance (Wasserstein) and a Gromov-Wasserstein distance (GW).

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.

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

TL;DR

Two distance measures to compare directed graphs based on variants of optimal transport based on variants of optimal transport(OT) are proposed: an earth movers distance (Wasserstein) and a Gromov-Wasserstein distance (GW).

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.
Paper Structure (13 sections, 14 equations, 1 figure, 1 table)

This paper contains 13 sections, 14 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Methods
  4. Node-to-Node distances for directed graphs
  5. Generalized Effective Resistance Distance (GRD)
  6. Hitting Time Based Distance (HTD)
  7. Optimal Transport distances for Directed Weighted Graphs
  8. Gromov-Wasserstein distance for directed graphs
  9. Wasserstein Distance for directed graphs
  10. Numerical Experiments
  11. Experiments on synthetic graphs: an illustrative example
  12. Case Study: single-cell RNAseq derived cell-cell interactions disease clustering
  13. Conclusions

Figures (1)

  • Figure 1: Optimal Transport distance between directed graphs. We consider an illustrative example network, consisting of four directed "local" cycles (indicated by node color), who are again connected "globally" in a cyclic way. Left: original reference network (note that the network is strongly connected). Middle: We perturb the network by swapping the direction of one of the "local" edges within one of the four cycles. Right: We perturb the network by swapping the direction of one of the "global" edges connecting the four cycles. The table below displays the obtained distances considering the following three baselines: the Frobenius norm of the difference between the adjacency matrices; the optimal transport-based GOT distance Maretic:2019, when considering the graph as undirected, and the optimal transport-based GWOT distance Chowdhury:2019 (again considering the graph as undirected). We contrast these with the results obtained via our Gromow-Wasserstein and Wasserstein (earth movers distance) formulation based on both generalized effective resistance and hitting time distance. Note that only our distance metrics that account for the directionality of the graph edges can distinguish these different cases.