Coupling and Tensorization of Kinetic Theory and Graph Theory
Datong Zhou
TL;DR
The paper develops a rigorous framework for the mean-field limit of non-exchangeable multi-agent systems by coupling kinetic theory with graphon theory through a convex bi-coupling distance. It introduces observables that tensorize agent laws with graph homomorphism densities, extending graph limit tools to functional (Hilbert-valued) graphons and proving stability and convergence to an extended Vlasov PDE. The main contributions include a discrete-to-continuum limit theorem, a Sobolev-based compactness approach, and a comprehensive tensorized BBGKY-type hierarchy that unifies mean-field and graph-limit perspectives for non-exchangeable dynamics. This work provides a principled pathway to analyze large-scale, heterogeneous networks where connection weights and agent identities influence dynamics, with potential implications for data-driven modeling and numerical computation in complex systems.
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.