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.
@article{arxiv.2601.04193,
title = {A discrete Benamou-Brenier formulation of Optimal Transport on graphs},
author = {Kieran Morris and Oliver Johnson},
journal= {arXiv preprint arXiv:2601.04193},
year = {2026}
}