Nonlocal cross-interaction systems on graphs: nonquadratic finslerian structure and nonlinear mobilities

We study the evolution of a system of two species with nonlinear mobility and nonlocal interactions on a graph whose vertices are given by an arbitrary, positive measure. To this end, we extend a recently introduced 2-Wasserstein-type quasi-metric on generalized graphs, which is based on an upwind-interpolation, to the case of two-species systems; concave, nonlinear mobilities; and p not equal 2. We provide a rigorous interpretation of the interaction system as a gradient flow in the Finslerian setting, arising from the new quasi-metric.