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Vlasov Equations on Directed Hypergraph Measures

Christian Kuehn, Chuang Xu

TL;DR

This work develops a measure-theoretic mean-field framework for interacting particle systems on directed hypergraphs by introducing directed hypergraph measures (DHGMs). It establishes well-posedness of the Vlasov equation on DHGMs and proves that empirical distributions of large networks with higher-order interactions converge to this Vlasov limit, via a fiberized-characteristics approach extended to higher dimensions. The framework is applied to Kuramoto-Sakaguchi, epidemic, and Lotka-Volterra models with higher-order couplings, demonstrating the broad applicability to physics, biology, and ecology. The results provide a rigorous bridge between finite multi-layer hypergraph dynamics and their continuum mean-field limits, enabling rigorous analysis and approximation of complex systems with higher-order interactions.

Abstract

In this paper we propose a framework to investigate the mean field limit (MFL) of interacting particle systems on directed hypergraphs. We provide a non-trivial measure-theoretic viewpoint and make extensions of directed hypergraphs as directed hypergraph measures (DHGMs), which are measure-valued functions on a compact metric space. These DHGMs can be regarded as hypergraph limits which include limits of a sequence of hypergraphs that are sparse, dense, or of intermediate densities. Our main results show that the Vlasov equation on DHGMs are well-posed and its solution can be approximated by empirical distributions of large networks of higher-order interactions. The results are applied to a Kuramoto network in physics, an epidemic network, and an ecological network, all of which include higher-order interactions. To prove the main results on the approximation and well-posedness of the Vlasov equation on DHGMs, we robustly generalize the method of [Kuehn, Xu. Vlasov equations on digraph measures, JDE, 339 (2022), 261--349] to higher-dimensions.

Vlasov Equations on Directed Hypergraph Measures

TL;DR

This work develops a measure-theoretic mean-field framework for interacting particle systems on directed hypergraphs by introducing directed hypergraph measures (DHGMs). It establishes well-posedness of the Vlasov equation on DHGMs and proves that empirical distributions of large networks with higher-order interactions converge to this Vlasov limit, via a fiberized-characteristics approach extended to higher dimensions. The framework is applied to Kuramoto-Sakaguchi, epidemic, and Lotka-Volterra models with higher-order couplings, demonstrating the broad applicability to physics, biology, and ecology. The results provide a rigorous bridge between finite multi-layer hypergraph dynamics and their continuum mean-field limits, enabling rigorous analysis and approximation of complex systems with higher-order interactions.

Abstract

In this paper we propose a framework to investigate the mean field limit (MFL) of interacting particle systems on directed hypergraphs. We provide a non-trivial measure-theoretic viewpoint and make extensions of directed hypergraphs as directed hypergraph measures (DHGMs), which are measure-valued functions on a compact metric space. These DHGMs can be regarded as hypergraph limits which include limits of a sequence of hypergraphs that are sparse, dense, or of intermediate densities. Our main results show that the Vlasov equation on DHGMs are well-posed and its solution can be approximated by empirical distributions of large networks of higher-order interactions. The results are applied to a Kuramoto network in physics, an epidemic network, and an ecological network, all of which include higher-order interactions. To prove the main results on the approximation and well-posedness of the Vlasov equation on DHGMs, we robustly generalize the method of [Kuehn, Xu. Vlasov equations on digraph measures, JDE, 339 (2022), 261--349] to higher-dimensions.
Paper Structure (15 sections, 17 theorems, 146 equations, 6 figures)

This paper contains 15 sections, 17 theorems, 146 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyper-digraph measures
  4. Generalized interacting particle system
  5. Assumptions
  6. Well-posedness of the Vlasov equation
  7. Approximation of time-dependent solutions to VE
  8. Applications
  9. A Kuramoto-Sakaguchi model with heterogeneous higher order interactions
  10. An epidemic model with higher-order interactions
  11. Lotka-Volterra model with dispersal on a hypergraph
  12. Discussion
  13. Proof of Theorem \ref{['th-approx']}
  14. Proof of Proposition \ref{['prop-continuousdependence']}
  15. Proof of Proposition \ref{['prop-sol-fixedpoint']}

Key Result

Proposition 2.3

For $i=1,2$, let $X_i$ be closed subsets of a finite dimensional Euclidean space. Assume $X_1$ and $X_2$ are both compact, and $(X_1,\mathfrak{B}(X_1),\mu_{X_1})$ is a probability space. Let $\mathcal{I}\subseteq\mathbb{R}$ be a compact interval.

Figures (6)

  • Figure 1: A directed hypergraph of $4$ vertices $\{1,2,3,4\}$ and rank $4$, two directed hyper-edges of cardinality 2 ($(1,3)$ and $(3,1)$), two directed hyper-edges of cardinality 3 ($(4,2,1)$ and $(2,3,4)$), and one hyper-edge of cardinality 4 ($(1,4,3,2)$).
  • Figure 2: Venn diagram for different hypergraph limits.
  • Figure 3: Example \ref{['ex-torical-graphop']}. Torical graph measure. Every point on the blue circle connects to every point on the red circle.
  • Figure 4: $3$-uniform triangular hypergraph $\mathcal{H}^5$ in Example \ref{['ex-triangle']}. A typical hyper-edge consisting of three vertices of an equilateral triangle of length 5 is colored in green (all others are colored in blue). The shaded area of a darker color consists of all vertices which belong to one "triangle" hyperedge located in the lower left corner.
  • Figure 5: Example \ref{['ex-1']} for $k=3$.
  • ...and 1 more figures

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Remark 2.7
  • Example 2.8
  • Definition 2.9
  • ...and 41 more