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Large-population limits of non-exchangeable particle systems

Nathalie Ayi, Nastassia Pouradier Duteil

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.

Large-population limits of non-exchangeable particle systems

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.
Paper Structure (46 sections, 1 theorem, 146 equations, 5 figures)

This paper contains 46 sections, 1 theorem, 146 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Mean-field limit of exchangeable particle systems.
  4. Exchangeable vs. non-exchangeable particle systems.
  5. Why the classical mean-field limit fails for non-exchangeable particle systems.
  6. New results in Limits of Graphs.
  7. The non-exchangeable mean-field limit and the continuum limit.
  8. Adaptive dynamical networks.
  9. Notations and preliminary remarks
  10. Large-population limits of non-exchangeable particle systems on static graphs
  11. Mean-field limit
  12. Dense graphs
  13. Lipschitz graphon.
  14. Non-Lipschitz graphon.
  15. General interaction function.
  16. ...and 31 more sections

Key Result

Proposition 1

The set of measures is dense in $\mathcal{P}(\mathbb{R}^d)$ for the weak topology. Moreover, for if $d=1$, for any $\mu_0\in\mathcal{P}(\mathbb{R})$, there exists a measurable function $x_0:I\rightarrow \mathbb{R}$ such that $\int_I \delta_{x_0(\xi)}(x) d\xi$.

Figures (5)

  • Figure 1: Graph and adjacency matrix associated with System \ref{['eq:Kuramoto_intro_2']}, for $N=10$ and $k = 2$.
  • Figure 2: Left and center: Pixel matrices of the graphs associated with \ref{['eq:Kuramoto_intro']} for $N=10$ and $N=50$, with $k = \frac{N}{5}$. Right: Plot of the limit graphon $w$.
  • Figure 3: Links between the different equations. The red arrows show the large-population limits described in Sections \ref{['sec:Graphs_MFL']} and \ref{['sec:Graphs_GL']}. The dashed arrows 1, 2 and 3 are explained in Sections \ref{['sec:Cont2NEMFL']}, \ref{['non_exchangeable_to_graph_limit']} and \ref{['sec:subordination']}. Arrow 4 corresponds to Remark \ref{['rem:arrow4']}.
  • Figure 4: Evolving-weighted graph associated to \ref{['eq:syst-gen']} in the case of three agents.
  • Figure 5: Links between the different equations. The arrow 1 corresponds to the weight dynamics $\psi_1(\xi,x(t,\cdot),m(t,\cdot)) = \int_{I\times\mathbb{R}^d \times \mathbb{R}_+^*} \tilde{S}(\xi,x(t,\xi),m(t,\xi),\zeta,y,n) d\tilde{\mu}_t^\zeta(y,n)d\zeta$, the arrow 2 to the weight dynamics $\psi_2(\xi,x(t,\cdot),m(t,\cdot)) = \int_{\mathbb{R}^d \times \mathbb{R}_+^*} \overline{S}(x(t,\xi),m(t,\xi),y,n) d\overline{\mu}_t(y,n)$ and the arrow 3 to the weight dynamics $\psi_3(\xi,x(t,\cdot),m(t,\cdot)) = m(t,\xi) \int_{\mathbb{R}^d } \hat{S}(x(t,\xi),y) d\hat{\mu}_t(y)$. The arrows 4 and 5 correspond to Remark \ref{['arrow4_5']}.

Theorems & Definitions (20)

  • Definition 1
  • Remark 1.1
  • Definition 2
  • Remark 2.1
  • Definition 3
  • Remark 2.2
  • Remark 2.3
  • Definition 4
  • Definition 5
  • Remark 2.4
  • ...and 10 more