  We consider dynamics driven by interaction energies on graphs. We introduce
graph analogues of the continuum nonlocal-interaction equation and interpret
them as gradient flows with respect to a graph Wasserstein distance. The
particular Wasserstein distance we consider arises from the graph analogue of
the Benamou-Brenier formulation where the graph continuity equation uses an
upwind interpolation to define the density along the edges. While this approach
has both theoretical and computational advantages, the resulting distance is
only a quasi-metric. We investigate this quasi-metric both on graphs and on
more general structures where the set of "vertices" is an arbitrary positive
measure. We call the resulting gradient flow of the nonlocal-interaction energy
the nonlocal nonlocal-interaction equation (NL$^2$IE). We develop the existence
theory for the solutions of the NL$^2$IE as curves of maximal slope with
respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the
solutions of the NL$^2$IE on graphs converge as the empirical measures of the
set of vertices converge weakly, which establishes a valuable
discrete-to-continuum convergence result.
