  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.
