Non-Markovian heat flows on directed hypergraphs

2025/10/20 by Mugnolo, Delio
#05C50 #05C65 #39A12 #47D06 #Analysis of PDEs (math.AP) #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.17497

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.

Citations

• Discrete Heat Kernels on Simplicial Complexes and Its Application to Functional Brain Networks
• On Hodge Laplacians on General Simplicial Complexes
• Non-Positivity of the heat equation with non-local Robin boundary conditions
• Calculus of variations on hypergraphs
• p-Laplacian Operators on Hypergraphs
• Heat equation on the hypergraph containing vertices with given data
• Geometry of nonequilibrium reaction networks
• Evolution equations with eventually positive solutions
• Nonlinear Evolution Equation Associated with Hypergraph Laplacian
• Random walks and Laplacians on hypergraphs: When do they match?
• What Are Higher-Order Networks?
• Node and Edge Nonlinear Eigenvector Centrality for Hypergraphs
• Graph Classification via Heat Diffusion on Simplicial Complexes
• Eventual Domination for Linear Evolution Equations
• Hypergraph Neural Networks
• Finding Cheeger Cuts in Hypergraphs via Heat Equation
• Cheeger Inequalities for Submodular Transformations
• Hypergraph Markov Operators, Eigenvalues and Approximation Algorithms
• The Total Variation on Hypergraphs - Learning on Hypergraphs Revisited
• On spectral hypergraph theory of the adjacency tensor
• Interlacing inequalities for eigenvalues of discrete Laplace operators
• Spectra of uniform hypergraphs
• Spectra of combinatorial Laplace operators on simplicial complexes
• Spectra of Hypergraphs and Applications

Related

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• Spectral Properties of Laplacian and Stochastic Diffusion Process for Edge Expansion in Hypergraphs
• Heat equation on the hypergraph containing vertices with given data
• Diffusion Operator and Spectral Analysis for Directed Hypergraph Laplacian