Large-population limits of non-exchangeable particle systems

TL;DR

This paper reviews new methods for analyzing the behavior of large non-exchangeable particle systems, where particle identities matter, by tracking individual identities and deriving limit equations.

Contribution

It introduces novel approaches for the large-population limit of non-exchangeable systems, focusing on particle identities and their evolution.

Findings

Non-exchangeable mean-field limit describes particle density evolution.

Continuum limit provides equations for individual particle trajectories.

Applicable to static and adaptive network frameworks.

Abstract

A particle system is said to be non-exchangeable if two particles cannot be exchanged without modifying the overall dynamics. Because of this property, the classical mean-field approach fails to provide a limit equation when the number of particles tends to infinity. In this review, we present novel approaches for the large-population limit of non-exchangeable particle systems, based on the idea of keeping track of the identities of the particles. These can be classified in two categories. The non-exchangeable mean-field limit describes the evolution of the particle density on the product space of particle positions and labels. Instead, the continuum limit allows to obtain an equation for the evolution of each particle's position as a function of its (continuous) label. We expose each of these approaches in the frameworks of static and adaptive networks.