Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Datong Zhou

TL;DR

This work develops a non-exchangeable mean-field framework for large systems in which both particle states and evolving interaction weights co-determine the dynamics. By leveraging an Aldous–Hoover representation on a filtered latent space and introducing the delta_{W_1,\square} metric, the authors prove propagation of dissociatedness and an empirical stability bound that jointly capture state and weight structures. A graphon-inspired Sampling Lemma is established to quantify finite-sample approximation errors, linking the classical chaos paradigm to a global, structure-aware limit. The results provide a rigorous, quantitative platform for analyzing co-evolving networks with Hebbian-like plasticity and dynamic interaction topologies, with potential applications to neural systems and other complex adaptive networks.

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.

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

TL;DR

This work develops a non-exchangeable mean-field framework for large systems in which both particle states and evolving interaction weights co-determine the dynamics. By leveraging an Aldous–Hoover representation on a filtered latent space and introducing the delta_{W_1,\square} metric, the authors prove propagation of dissociatedness and an empirical stability bound that jointly capture state and weight structures. A graphon-inspired Sampling Lemma is established to quantify finite-sample approximation errors, linking the classical chaos paradigm to a global, structure-aware limit. The results provide a rigorous, quantitative platform for analyzing co-evolving networks with Hebbian-like plasticity and dynamic interaction topologies, with potential applications to neural systems and other complex adaptive networks.

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.

Paper Structure

This paper contains 27 sections, 9 theorems, 126 equations.

Table of Contents

  1. Overview
  2. The paradigm of mean-field limits: from exchangeable to non-exchangeable, and plasticity
  3. A prototypical model
  4. From classical chaos to propagation of dissociatedness
  5. Empirical stability
  6. Organization of the article
  7. Discussion and Future Perspectives
  8. Graph kernel mean-field limit and the fiberwise perspective
  9. Dynamic random networks and probabilistic foundations
  10. Co-evolutionary dynamics and limit characterizations
  11. Relation to the hierarchical approach
  12. Future perspectives
  13. Sparser interaction weights
  14. Multi-body interactions and hypergraphs
  15. Diverse dynamics for particles and weights
  16. ...and 12 more sections

Key Result

Theorem 1.2

Assume that the coefficients of the system eqn:main_SDEs satisfy the regularity conditions in Assumption assu:coefficients. Consider initial data $(\bm{X}_{(0)}, \bm{W}_{(0)})$ for a system of size $N$ that satisfies the dissociatedness property (Definition defi:dissociatedness) and the moment bound

Theorems & Definitions (27)

  • Definition 1.1: Dissociatedness
  • Theorem 1.2: Propagation of Dissociatedness
  • Proposition 1.3: Aldous-Hoover Representation
  • Theorem 1.4: Empirical Stability of State-Weight Structure
  • Lemma 1.5: Sampling Lemma
  • Conjecture 2.1: Mean-field weight noise does not affect the limit
  • Conjecture 2.2: Partial homogenization
  • Remark 3.1: Representative Dynamics vs. Continuum of Agents
  • Remark 3.3: On the Integral Separable Structure
  • Proposition 3.5: Well-posedness and Stability of McKean-Vlasov SDEs
  • ...and 17 more