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

arXiv: https://arxiv.org/abs/2309.07030 · HTML: https://mlsanity.com/paper/2309.07030/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 21 references that are in the map.
- First posted on arXiv 13 Sep 2023.

## Builds on (most cited first, 7 of 7)

- [Gromov-Wasserstein Learning for Graph Matching and Node Embedding](https://mlsanity.com/paper/1901.06003/) (2019)
- [GOT: An Optimal Transport framework for Graph comparison](https://mlsanity.com/paper/1906.02085/) (2019)
- [PageRank beyond the Web](https://mlsanity.com/paper/1407.5107/) (2014)
- [A New Notion of Effective Resistance for Directed Graphs-Part I: Definition and Properties](https://mlsanity.com/paper/1310.5163/) (2013)
- [A New Notion of Effective Resistance for Directed Graphs-Part II: Computing Resistances](https://mlsanity.com/paper/1310.5168/) (2013)
- [A metric on directed graphs and Markov chains based on hitting probabilities](https://mlsanity.com/paper/2006.14482/) (2020)
- [Comparing directed networks via denoising graphlet distributions](https://mlsanity.com/paper/2207.09827/) (2022)

## Abstract (arXiv)

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.