Abstract
The talk focuses on dynamics driven by interaction energies on graphs. More precisely, in a recent paper we introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance, using Benamou--Brenier formulation. 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 (NLNLIE), and develop existence theory as curve of maximal slope. Furthermore, we establish a discrete-to-continuum convergence result with respect to the number of vertices. Finally, by means of a fixed-point argument we can show existence and uniqueness of strong solutions for some interpolations. The talk is based on works in collaboration with F. S. Patacchini (IFP Energies Nouvelles), A. Schlichting (University of Münster), and D. Slepcev (Carnegie Mellon University).