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Interacting particle systems on sparse $W$-random graphs

Carla Crucianelli, Ludovic Tangpi

Abstract

We consider a general interacting particle system with interactions on a random graph, and study the large population limit of this system. When the sequence of underlying graphs converges to a graphon, we show convergence of the interacting particle system to a so called graphon stochastic differential equation. This is a system of uncountable many SDEs of McKean-Vlasov type driven by a continuum of Brownian motions. We make sense of this equation in a way that retains joint measurability and essentially pairwise independence of the driving Brownian motions of the system by using the framework of Fubini extension. The convergence results are general enough to cover nonlinear interactions, as well as various examples of sparse graphs. A crucial idea is to work with unbounded graphons and use the $L^p$ theory of sparse graph convergence.

Interacting particle systems on sparse $W$-random graphs

Abstract

We consider a general interacting particle system with interactions on a random graph, and study the large population limit of this system. When the sequence of underlying graphs converges to a graphon, we show convergence of the interacting particle system to a so called graphon stochastic differential equation. This is a system of uncountable many SDEs of McKean-Vlasov type driven by a continuum of Brownian motions. We make sense of this equation in a way that retains joint measurability and essentially pairwise independence of the driving Brownian motions of the system by using the framework of Fubini extension. The convergence results are general enough to cover nonlinear interactions, as well as various examples of sparse graphs. A crucial idea is to work with unbounded graphons and use the theory of sparse graph convergence.

Paper Structure

This paper contains 21 sections, 35 theorems, 217 equations.

Table of Contents

  1. Introduction and main results
  2. Probabilistic setting and general graphon SDEs
  3. Main results of the paper
  4. Well-posedness
  5. Propagation of chaos
  6. Literature review
  7. Outline of the rest of the paper
  8. Frequently used notation
  9. Existence, uniqueness and stability of graphon SDEs
  10. Measure theoretic preliminaries
  11. Measurability of SDEs depending on a parameter
  12. Proof of Theorem \ref{['thm:existence-uniqueness']}
  13. Stability of graphon SDEs
  14. Propagation of chaos and related convergence results
  15. Stability and moment estimates
  16. ...and 6 more sections

Key Result

Theorem 1.4

sun2008individual Let $I'$ be the unit interval and $E$ a Polish space. There exists a probability space $(I',\mathcal{I'},\lambda')$ that is an extension of the Lebesgue measure on $I'$, a probability space $(\Omega', \mathcal{F}',\mathbb{P}')$ and a Fubini extension $(I'\times\Omega', \mathcal{I'}

Theorems & Definitions (76)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.6
  • Definition 1.7
  • Theorem 1.9
  • Remark 1.10
  • Remark 1.12
  • Theorem 1.15
  • ...and 66 more