Non-exchangeable mean-field theory for adaptive weights: propagation of dissociatedness and graphon sampling lemma

Abstract

We develop a mean-field theory for large, non-exchangeable particle (agent) systems where the states and interaction weights co-evolve in a coupled system of SDEs. A first main result is the establishment of the propagation of dissociatedness, a conceptual generalization of the classical propagation of chaos that accommodates the intrinsic local correlations between particles and their weights. The limiting McKean-Vlasov process is characterized by an Aldous-Hoover representation on a filtered probability space, beyond the standard one-particle law (or a family thereof). Paralleling the classical equivalence between propagation of chaos and the convergence of empirical measures to the one-particle law, we show that the propagation of dissociatedness corresponds to the convergence of the empirical structure under a distance unifying the Wasserstein distance for particles and the cut distance for weights. This quantitative stability is grounded in an adaptation of the sampling lemma from dense graph theory, analogous to the classical concentration results for empirical measures in the Wasserstein distance.