# A discrete Benamou-Brenier formulation of Optimal Transport on graphs

arXiv: https://arxiv.org/abs/2601.04193 · HTML: https://mlsanity.com/paper/2601.04193/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 21 references that are in the map.
- First posted on arXiv 7 Jan 2026.

## Builds on (most cited first, 4 of 4)

- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [Large-Scale Wasserstein Gradient Flows](https://mlsanity.com/paper/2106.00736/) (2021)
- [Wasserstein Dependency Measure for Representation Learning](https://mlsanity.com/paper/1903.11780/) (2019)

## Abstract (arXiv)

We propose a discrete transport equation on graphs which connects distributions on both vertices and edges. We then derive a discrete analogue of the Benamou-Brenier formulation for Wasserstein-1 distance on a graph and as a result classify all W1 geodesics on graphs.