Non-Markovian heat flows on directed hypergraphs

TL;DR

This paper develops a new framework for heat flows on directed hypergraphs, revealing complex non-Markovian diffusion behaviors and eigenvalue bounds, with implications for understanding large-time dynamics.

Contribution

Introduces a semigroup framework for Laplacians on directed hypergraphs, extending classical models and analyzing non-Markovian diffusion phenomena.

Findings

Heat flows can lose positivity and $ abla$-contractivity but may recover asymptotically.

Eigenvalue bounds describe large-time behavior of heat flow.

Examples illustrate phenomena on hypergraphs derived from oriented graphs and the Fano plane.

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.