Continuous-Flow Graph Transportation Distances

Optimal transportation distances are valuable for comparing and analyzing probability distributions, but larger-scale computational techniques for the theoretically favorable quadratic case are limited to smooth domains or regularized approximations. Motivated by fluid flowbased transportation on Rn, however, this paper introduces an alternative definition of optimal transportation between distributions over graph vertices. This new distance still satisfies the triangle inequality but has better scaling and a connection to continuous theories of transportation. It is constructed by adapting a Riemannian structure over probability distributions to the graph case, providing transportation distances as shortest-paths in probability space. After defining and analyzing theoretical properties of our new distance, we provide a time discretization as well as experiments verifying its effectiveness.