Mean-field limits for interacting particle systems on general adaptive dynamical networks

TL;DR

This paper investigates the large-population behavior of agents interacting on evolving networks, deriving a Vlasov-type equation to describe the system's asymptotics and establishing connections between mean-field and graph limit approaches.

Contribution

It introduces a unified framework for analyzing the mean-field limits of particle systems on adaptive networks, including new methods to handle general interaction structures.

Findings

Derived a Vlasov-type equation for large populations on adaptive networks.

Established well-posedness and stability of the limiting equation.

Connected mean-field and graph limit descriptions of the system.

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.