The fields of effective resistance and optimal transport on graphs are filled with rich connections to combinatorics, geometry, machine learning, and beyond. In this article we put forth a bold claim: that the two fields should be understood as one and the same, up to a choice of $p$. We make this claim precise by introducing the parameterized family of $p$-Beckmann distances for probability measures on graphs and relate them sharply to certain Wasserstein distances. Then, we break open a suite of results including explicit connections to optimal stopping times and random walks on graphs, graph Sobolev spaces, and a Benamou-Brenier type formula for $2$-Beckmann distance. We further explore empirical implications in the world of unsupervised learning for graph data and propose further study of the usage of these metrics where Wasserstein distance may produce computational bottlenecks.

图上的有效电阻和最优传输领域与组合数学、几何学、机器学习等具有广泛关联。本文提出了一个大胆的观点，即这两个领域应被理解为相同的，仅在于选择$p$。通过引入参数化的$p$-Beckmann距离家族来确立这一观点，并将其与某些Wasserstein距离紧密关联。进而，我们揭示了一系列结果，包括与图上最优停时、随机行走、图Sobolev空间以及$2$-Beckmann距离的Benamou-Brenier公式的明确联系。我们还进一步探讨了在图数据的无监督学习领域中的实证意义，并建议进一步研究在计算中Wasserstein距离可能存在的瓶颈问题。

抵抗力是关键：图上有效抵抗力与某些最优输运问题的等价性