Mean-field and graph limits for collective dynamics models with time-varying weights

TL;DR

This paper analyzes opinion dynamics models with time-varying influence weights, establishing rigorous connections between graph and mean-field limits, and providing numerical illustrations of these theoretical results.

Contribution

It introduces a unified framework for deriving mean-field and graph limits in models with evolving influence weights, including new proofs and general conditions.

Findings

Existence and uniqueness of solutions for the models.

Rigorous justification of the graph limit in a general setting.

Demonstration of the subordination of the mean-field limit to the graph limit.

Abstract

In this paper, we study a model for opinion dynamics where the influence weights of agents evolve in time via an equation which is coupled with the opinions' evolution. We explore the natural question of the large population limit with two approaches: the now classical mean-field limit and the more recent graph limit. After establishing the existence and uniqueness of solutions to the models that we will consider, we provide a rigorous mathematical justification for taking the graph limit in a general context. Then, establishing the key notion of indistinguishability, which is a necessary framework to consider the mean-field limit, we prove the subordination of the mean-field limit to the graph one in that context. This actually provides an alternative (but weaker) proof for the mean-field limit. We conclude by showing some numerical simulations to illustrate our results.