Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

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