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Mean-field limits for interacting particles on general adaptive dynamical networks

Nathalie Ayi

TL;DR

The paper develops a rigorous mean-field theory for interacting particle systems on general adaptive dynamical networks, where the graph structure evolves with the agents’ states and is non-exchangeable. It derives a closed Vlasov-type equation on an extended phase space that includes identities and edge weights, establishing two complementary routes: a Sznitman-style probabilistic mean-field limit under a weight-independence assumption and a deterministic graph-limit approach that relaxes this constraint. The authors prove well-posedness and stability for the limiting equation, quantify propagation of independence, and connect the continuum-graph limit with the mean-field limit, thereby providing a unified description of the asymptotic dynamics. The framework accommodates higher-order, edge–edge interactions through the weight dynamics and yields quantitative convergence results, highlighting the impact of adaptive connectivity on collective behavior in opinion-like models.

Abstract

We study the large-population limit of interacting particle systems evolving on adaptive dynamical networks, motivated in particular by models of opinion dynamics. In such systems, agents interact through weighted graphs whose structure evolves over time in a coupled manner with the agents' states, leading to non-exchangeable dynamics. In the dense-graph regime, we show that the asymptotic behavior is described by a Vlasov-type equation posed on an extended phase space that includes both the agents' states and identities and the evolving interaction weights. We establish this limiting equation through two complementary approaches. The first follows the mean-field methodology in the spirit of Sznitman [28]. In this framework, we impose the additional assumption that the weight dynamics is independent of one of the agent's states, an assumption that remains well motivated from a modeling perspective and allows for a direct derivation of the mean-field limit. The second approach is based on the graph limit framework and is formulated in a deterministic setting. This perspective makes it possible to remove the aforementioned restriction on the weight dynamics and to handle more general interaction structures. Our analysis includes wellposedness and stability results for the limiting Vlasov-type equation, as well as quantitative estimates ensuring the propagation of independence. We further clarify the relationship between the continuum (graph limit) formulation and the mean-field limit, thereby providing a unified description of the asymptotic dynamics of interacting particle systems on adaptive dynamical networks.

Mean-field limits for interacting particles on general adaptive dynamical networks

TL;DR

The paper develops a rigorous mean-field theory for interacting particle systems on general adaptive dynamical networks, where the graph structure evolves with the agents’ states and is non-exchangeable. It derives a closed Vlasov-type equation on an extended phase space that includes identities and edge weights, establishing two complementary routes: a Sznitman-style probabilistic mean-field limit under a weight-independence assumption and a deterministic graph-limit approach that relaxes this constraint. The authors prove well-posedness and stability for the limiting equation, quantify propagation of independence, and connect the continuum-graph limit with the mean-field limit, thereby providing a unified description of the asymptotic dynamics. The framework accommodates higher-order, edge–edge interactions through the weight dynamics and yields quantitative convergence results, highlighting the impact of adaptive connectivity on collective behavior in opinion-like models.

Abstract

We study the large-population limit of interacting particle systems evolving on adaptive dynamical networks, motivated in particular by models of opinion dynamics. In such systems, agents interact through weighted graphs whose structure evolves over time in a coupled manner with the agents' states, leading to non-exchangeable dynamics. In the dense-graph regime, we show that the asymptotic behavior is described by a Vlasov-type equation posed on an extended phase space that includes both the agents' states and identities and the evolving interaction weights. We establish this limiting equation through two complementary approaches. The first follows the mean-field methodology in the spirit of Sznitman [28]. In this framework, we impose the additional assumption that the weight dynamics is independent of one of the agent's states, an assumption that remains well motivated from a modeling perspective and allows for a direct derivation of the mean-field limit. The second approach is based on the graph limit framework and is formulated in a deterministic setting. This perspective makes it possible to remove the aforementioned restriction on the weight dynamics and to handle more general interaction structures. Our analysis includes wellposedness and stability results for the limiting Vlasov-type equation, as well as quantitative estimates ensuring the propagation of independence. We further clarify the relationship between the continuum (graph limit) formulation and the mean-field limit, thereby providing a unified description of the asymptotic dynamics of interacting particle systems on adaptive dynamical networks.
Paper Structure (16 sections, 23 theorems, 174 equations)

This paper contains 16 sections, 23 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Presentation of the model and main results
  3. Setting
  4. Functional spaces
  5. Statement of the results
  6. Well-posedness of the Vlasov-type equation
  7. Well-posedness of the characteristics
  8. Well posedness of the Vlasov-type equation
  9. Stability estimates for the Vlasov-type equation
  10. Propagation of independance
  11. An intermediate particle system
  12. Scaling assumptions for the graph
  13. Control of the error
  14. Comparison of the graphon reformulations
  15. Mean-field limit
  16. ...and 1 more sections

Key Result

Theorem 1

Consider any $\nu$ in $\mathcal{P}(I)$ and $\mu \in \mathcal{P}_\nu( I^2 \times \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R})$. Then, there exists an almost everywhere uniquely defined Borel family $(\mu^{\xi,\zeta})_{\xi,\zeta \in I} \subset \mathcal{P}(\mathbb{R}^d \times \mathbb{R}^d \times for every bounded Borel-measure map $\varphi : I^2 \times \mathbb{R}^d \times \mathbb{R}^d \times

Theorems & Definitions (60)

  • Remark 1.1
  • Remark 1.2
  • Remark 2.1
  • Remark 2.2
  • Definition 1: Borel family of probability measures
  • Definition 2: Fibered probability measures
  • Theorem 1: Disintegration theorem
  • Definition 3
  • Remark 2.3
  • Theorem 2: Time-dependent disintegrations
  • ...and 50 more