Nonlocalinteraction equation on graphs: gradient flow structure and continuum limit
Abstract
We consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocalinteraction 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 BenamouBrenier 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 quasimetric. We investigate this quasimetric 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 nonlocalinteraction energy the nonlocal nonlocalinteraction 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 quasimetric. 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 discretetocontinuum convergence result.
 Publication:

arXiv eprints
 Pub Date:
 December 2019
 arXiv:
 arXiv:1912.09834
 Bibcode:
 2019arXiv191209834E
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Numerical Analysis;
 Mathematics  Probability;
 Mathematics  Statistics Theory
 EPrint:
 45 pages