Graph-to-local limit for a multi-species nonlocal cross-interaction system

TL;DR

This paper extends the analysis of nonlocal multi-species interaction systems on graphs, proving convergence to a gradient flow and establishing solution existence on tensor-weighted Euclidean spaces.

Contribution

It generalizes previous results to multiple species and introduces a tensor-weighted gradient flow framework for nonlocal interactions.

Findings

Proves mma-convergence of graph dynamics to a gradient flow.

Establishes existence of solutions on tensor-weighted Euclidean space.

Extends analysis to arbitrary number of species.

Abstract

In this note we continue the study of nonlocal interaction dynamics on a sequence of infinite graphs, extending the results of [Esposito et. al 2023+] to an arbitrary number of species. Our analysis relies on the observation that the graph dynamics form a gradient flow with respect to a non-symmetric Finslerian gradient structure. Keeping the nonlocal interaction energy fixed, while localising the graph structure, we are able to prove evolutionary {\Gamma}-convergence to an Otto-Wassertein-type gradient flow with a tensor-weighted, yet symmetric, inner product. As a byproduct this implies the existence of solutions to the multi-species non-local (cross-)interacation system on the tensor-weighted Euclidean space