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 $W_1$ geodesics on graphs.

翻译：我们提出一种图上的离散输运方程，该方程连接了顶点和边上的分布。随后，我们推导出图上Wasserstein-$1$距离的Benamou-Brenier公式的离散类比，并由此分类了图上的所有$W_1$测地线。