  In this paper, we address the numerical solution of the Optimal Transport
Problem on undirected weighted graphs, taking the shortest path distance as
transport cost. The optimal solution is obtained from the long-time limit of
the gradient descent dynamics. Among different time stepping procedures for the
discretization of this dynamics, a backward Euler time stepping scheme combined
with the inexact Newton-Raphson method results in a robust and accurate
approach for the solution of the Optimal Transport Problem on graphs. It is
found experimentally that the algorithm requires solving between
$\mathcal{O}(1)$ and $\mathcal{O}(M^{0.36})$ linear systems involving weighted
Laplacian matrices, where $M$ is the number of edges. These linear systems are
solved via algebraic multigrid methods, resulting in an efficient solver for
the Optimal Transport Problem on graphs.
