Coupling and Tensorization of Kinetic Theory and Graph Theory

TL;DR

This paper establishes a rigorous connection between multi-agent systems, mean-field limits, and graph theory by deriving a deterministic PDE limit using a novel bi-coupling distance, revealing deep links between these fields.

Contribution

It introduces a new framework combining kinetic theory and graph theory to analyze non-exchangeable multi-agent systems and proves convergence to a Vlasov PDE using a bi-coupling distance.

Findings

Convergence of large-scale dynamics to a deterministic limit.

Introduction of a bi-coupling distance for measure comparison.

Establishment of a relationship between mean-field and graph limiting theories.

Abstract

We study a non-exchangeable multi-agent system and rigorously derive a strong form of the mean-field limit. The convergence of the connection weights and the initial data implies convergence of large-scale dynamics toward a deterministic limit given by the corresponding extended Vlasov PDE, at any later time and any realization of randomness. This is established on what we call a bi-coupling distance defined through a convex optimization problem, which is an interpolation of the optimal transport between measures and the fractional overlay between graphs. The proof relies on a quantitative stability estimate of the so-called observables, which are tensorizations of agent laws and graph homomorphism densities. This reveals a profound relationship between mean-field theory and graph limiting theory, intersecting in the study of non-exchangeable systems.