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Non-Markovian heat flows on directed hypergraphs

Delio Mugnolo

TL;DR

The paper develops a semigroup framework for Laplacians on directed hypergraphs, revealing that diffusion can be non-Markovian with potential loss of positivity or mass conservation. It analyzes spectral properties of the hypergraph Laplacian $\mathcal{L}=\mathcal{I}\mathcal{I}^\top$ and its dual $\mathcal{L}^*=\mathcal{I}^\top\mathcal{I}$, derives eigenvalue bounds via hypergraph perturbations, and studies how positivity, $\infty$-contractivity, and stochasticity depend on combinatorial structure. It introduces notions of eventual positivity and asymptotic $\infty$-contractivity, explores sub-hypergraphs with Dirichlet conditions, and illustrates these phenomena with duals of graphs and the Fano plane. The results illuminate how higher-order interactions govern long-time diffusion and provide a foundation for extending to nonlinear dynamics and physical models in non-equilibrium systems. These insights broaden the understanding of diffusion on complex incidence structures beyond classical graph theory, highlighting rich, structure-dependent behaviours and guiding future research in higher-order network dynamics.

Abstract

We introduce a semigroup framework for Laplacians on directed hypergraphs, extending the classical heat flow models on graphs and establishing hypergraphs as prototypical models for non-Markovian diffusion. We apply spectral surgery methods to derive eigenvalue bounds, thus describing large-time behaviour of the heat flow. Unlike on standard graphs, heat flows on directed hypergraphs may lose positivity and/or $\infty$-contractivity, yet can recover them eventually or asymptotically under specific combinatorial configurations: examples based on duals of oriented graph and realisations of the Fano plane illustrate these phenomena. Our approach combines combinatorial, order-theoretic and linear-algebraic methods.

Non-Markovian heat flows on directed hypergraphs

TL;DR

The paper develops a semigroup framework for Laplacians on directed hypergraphs, revealing that diffusion can be non-Markovian with potential loss of positivity or mass conservation. It analyzes spectral properties of the hypergraph Laplacian and its dual , derives eigenvalue bounds via hypergraph perturbations, and studies how positivity, -contractivity, and stochasticity depend on combinatorial structure. It introduces notions of eventual positivity and asymptotic -contractivity, explores sub-hypergraphs with Dirichlet conditions, and illustrates these phenomena with duals of graphs and the Fano plane. The results illuminate how higher-order interactions govern long-time diffusion and provide a foundation for extending to nonlinear dynamics and physical models in non-equilibrium systems. These insights broaden the understanding of diffusion on complex incidence structures beyond classical graph theory, highlighting rich, structure-dependent behaviours and guiding future research in higher-order network dynamics.

Abstract

We introduce a semigroup framework for Laplacians on directed hypergraphs, extending the classical heat flow models on graphs and establishing hypergraphs as prototypical models for non-Markovian diffusion. We apply spectral surgery methods to derive eigenvalue bounds, thus describing large-time behaviour of the heat flow. Unlike on standard graphs, heat flows on directed hypergraphs may lose positivity and/or -contractivity, yet can recover them eventually or asymptotically under specific combinatorial configurations: examples based on duals of oriented graph and realisations of the Fano plane illustrate these phenomena. Our approach combines combinatorial, order-theoretic and linear-algebraic methods.
Paper Structure (14 sections, 32 theorems, 74 equations, 3 figures)

This paper contains 14 sections, 32 theorems, 74 equations, 3 figures.

Table of Contents

  1. Introduction
  2. General properties of directed hypergraphs
  3. Spectral Theory
  4. Hypergraph perturbation of graphs
  5. Order theoretical aspects
  6. Positivity and eventual positivity
  7. $\infty$-contractivity and eventual $\infty$-contractivity
  8. Sub-hypergraphs and boundary conditions
  9. Examples
  10. Dual directed hypergraphs
  11. Directed hypergraphs dual to graphs
  12. Abstract simplicial complexes as directed hypergraphs
  13. An illustrative example: the Fano plane
  14. Conclusions and outlook

Key Result

Lemma 2.1

Let $\mathsf{H}=(\mathsf{V},\mathsf{E})$ be a directed hypergraph. Then In particular,

Figures (3)

  • Figure 1: Two hypergraphs consisting of three vertices and one hyperedge
  • Figure 2: The time evolution driven by the semigroups $(\mathrm{e}^{-t\mathcal{L}_\mathsf{H}})_{t\ge 0}$ (left) and $(\mathrm{e}^{-t\mathcal{L}_\mathsf{G}})_{t\ge 0}$ (right) in \ref{['exa:graphyper-compare']} for the initial condition $u_0=010^\top$. The green (resp., red, blue) curves depict the time evolution of $u(t,\mathsf{v})$ for $\mathsf{v}=\mathsf{v}_1$ (resp., $\mathsf{v}=\mathsf{v}_2$, $\mathsf{v}=\mathsf{v}_3$) and $t\ge 0$.
  • Figure 7: A drawing of the Fano plane

Theorems & Definitions (81)

  • Definition 1
  • Definition 2
  • Example 1
  • Definition 3
  • Definition 4
  • Definition 4
  • Remark 1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • ...and 71 more