Nonlocal-interaction equations on graphs and gradient flows in nonlocal Wasserstein metric
Dejan Slepcev (Carnegie Mellon University)
We consider transport equations on graphs, where mass is distributed over vertices and is transported along the edges. The first part of the talk will deal with the graph analogue of the Wasserstein distance, in the particular case where the notion of density along edges is inspired by the upwind numerical schemes. 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. In the second part of the talk we will interpret the nonlocal-interaction equation equations on graphs as gradient flows with respect to the graph-Wasserstain quasi-metric of the nonlocal-interaction energy. We show that for graphs representing data sampled from a manifold, the solutions of the nonlocal-interaction equations on graphs converge to solutions of an integral equation on the manifold.