Hypergraph p-Laplacians and Scale Spaces

Fazeny, Ariane; Burger, Martin; Tenbrinck, Daniel; Lukin, Kseniaa

Preprint

PUBDB-2023-06362

Hypergraph p-Laplacians and Scale Spaces

Fazeny, A. (Corresponding author)DESY* ; Tenbrinck, D. ; Lukin, K. ; Burger, M.DESY*

2023

[10.3204/PUBDB-2023-06362]

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Please use a persistent id in citations: doi:10.3204/PUBDB-2023-06362

Report No.: arXiv:2309.15419

Abstract: The aim of this paper is to revisit the definition of differential operators on hypergraphs, which are a natural extension of graphs in systems based on interactions beyond pairs. {In particular, we focus on the definition of Laplacian and $p$-Laplace operators for oriented and unoriented hypergraphs,} their basic properties, variational structure, and their scale spaces. We illustrate that diffusion equations on hypergraphs are possible models for different applications such as information flow on social networks or image processing. Moreover, the spectral analysis and scale spaces induced by these operators provide a potential method to further analyze complex {data} and their multiscale structure. The quest for spectral analysis and suitable scale spaces on hypergraphs motivates in particular a definition of differential operators with trivial first eigenfunction and thus more interpretable second eigenfunctions. This property is not automatically satisfied in existing definitions of hypergraph $p$-Laplacians and we hence provide a novel axiomatic approach that extends previous definitions and can be specialized to satisfy such (or other) desired properties.

Note: The funding should include SFB TR 154, Subproject C06, but it was not possible to find it in the mask.

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