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Mean-field limit of non-exchangeable systems

Pierre-Emmanuel Jabin, David Poyato, Juan Soler

TL;DR

The article develops a mean-field theory for non-exchangeable, non-identical agents on sparse networks by introducing extended graphons, objects that live in $L^ ext{∞}_ ext{ξ} ext{M}_ ext{ζ} \,igcap\, L^ ext{∞}_ ext{ζ} ext{M}_ ext{ξ}$. It constructs a hierarchy of tree-indexed observables ${ au}(T,w,f)$ that jointly encode network structure and initial data, and proves convergence of these observables to a continuum limit described by a Vlasov-type equation with kernel $w$. A generalized McKean SDE is developed to capture propagation of independence in this non-exchangeable setting, and a robust compactness framework is built around an extended graphon representation to identify the limit $(w,f^0)$ and show convergence of empirical measures to $ frac{1}{N} extstyle\sum_i ext{delta}_{X_i}(t)$ toward $\

Abstract

This paper deals with the derivation of the mean-field limit for multi-agent systems on a large class of sparse graphs. More specifically, the case of non-exchangeable multi-agent systems consisting of non-identical agents is addressed. The analysis does not only involve PDEs and stochastic analysis but also graph theory through a new concept of limits of sparse graphs (extended graphons) that reflect the structure of the connectivities in the network and has critical effects on the collective dynamics. In this article some of the main restrictive hypothesis in the previous literature on the connectivities between the agents (dense graphs) and the cooperation between them (symmetric interactions) are removed.

Mean-field limit of non-exchangeable systems

TL;DR

The article develops a mean-field theory for non-exchangeable, non-identical agents on sparse networks by introducing extended graphons, objects that live in . It constructs a hierarchy of tree-indexed observables that jointly encode network structure and initial data, and proves convergence of these observables to a continuum limit described by a Vlasov-type equation with kernel . A generalized McKean SDE is developed to capture propagation of independence in this non-exchangeable setting, and a robust compactness framework is built around an extended graphon representation to identify the limit and show convergence of empirical measures to toward $\

Abstract

This paper deals with the derivation of the mean-field limit for multi-agent systems on a large class of sparse graphs. More specifically, the case of non-exchangeable multi-agent systems consisting of non-identical agents is addressed. The analysis does not only involve PDEs and stochastic analysis but also graph theory through a new concept of limits of sparse graphs (extended graphons) that reflect the structure of the connectivities in the network and has critical effects on the collective dynamics. In this article some of the main restrictive hypothesis in the previous literature on the connectivities between the agents (dense graphs) and the cooperation between them (symmetric interactions) are removed.
Paper Structure (30 sections, 27 theorems, 389 equations, 3 figures)

This paper contains 30 sections, 27 theorems, 389 equations, 3 figures.

Table of Contents

  1. Introduction. Multi-agent systems on graphs
  2. Mean--field limits for identical weights
  3. The use of graphons for multi-agent systems
  4. Our new result
  5. Notations, basic definitions, examples and sketch of the proof
  6. Notations
  7. The space $L^\infty([0,\ 1],\ \mathcal{M}([0,\ 1])$
  8. The space $L^\infty_\xi \mathcal{M}_\zeta\cap L^\infty_\zeta \mathcal{M}_\xi$ and examples
  9. Further comments on our extended graphons
  10. Example of convergence
  11. Sketch of the proof
  12. A generalized McKean SDE and propagation of independence
  13. A new hierarchy
  14. Hierarchy of observables indexed by trees
  15. A graphon-like representation of the system and its hierarchy
  16. ...and 15 more sections

Key Result

Theorem 1.1

Assume that the interaction kernel $K$ belongs to $W^{1,1}\cap W^{1,\infty}(\mathop{\mathrm{\mathbb{R}}}\nolimits^d)$ and consider a sequence $(X_i(t))_{1\leq i\leq N}$, solving the multi-agent system eq1 for connection weights satisfying only meanfieldscaling-vanishingweights. Assume moreover that Then, there exist $w\in L^\infty_\xi \mathcal{M}_\zeta\cap L^\infty_\zeta \mathcal{M}_\xi$ and $f\i

Figures (3)

  • Figure 1: Comparison of the various graph limit theories.
  • Figure 2: Trees in $\mathop{\mathrm{\mathsf{T}}}\nolimits_3$
  • Figure 3: Words in $\mathop{\mathrm{\mathsf{W}}}\nolimits_0$, $\mathop{\mathrm{\mathsf{W}}}\nolimits_1$ and $\mathop{\mathrm{\mathsf{W}}}\nolimits_2$

Theorems & Definitions (63)

  • Theorem 1.1
  • Lemma 3.1: McKean SDE
  • Proposition 3.2: Propagation of independence
  • proof
  • Remark 3.3: Exchangeable case
  • Definition 4.1: Graphs and trees
  • Remark 4.2: Directed graphs and directed trees.
  • Definition 4.3: Observables
  • Definition 4.4: Empirical graphons
  • Definition 4.5
  • ...and 53 more