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：我们提出了一种图上的离散传输方程，该方程同时关联了顶点和边上的分布。随后，我们推导出图上Wasserstein-$1$距离的离散Benamou-Brenier表述，并以此对图上所有$W_1$测地线进行了分类。