Hypergraph $p$-Laplacians and Scale Spaces
Ariane Fazeny, Daniel Tenbrinck, Kseniia Lukin, Martin Burger
TL;DR
The work extends diffusion PDEs from graphs to hypergraphs by defining gradient-based and adjoint operators on oriented hypergraphs and, separately, gradient-based and averaging-based Laplacians on unoriented hypergraphs. It establishes a variational structure for the vertex $p$-Laplacian and develops diffusion models for information flow in social networks as well as local and nonlocal image processing via scale spaces. Key contributions include generalized vertex $p$-Laplacians with flexible weighting, two complementary Laplacian constructions for unoriented hypergraphs, and comprehensive numerical demonstrations highlighting the benefits of hypergraph-based diffusion and spectral insights. The framework lays a foundation for hypergraph neural networks and further PDE/spectral analyses in higher-order networked data.
Abstract
This paper introduces gradient, adjoint, and $p$-Laplacian definitions for oriented hypergraphs as well as differential and averaging operators for unoriented hypergraphs. These definitions are used to define gradient flows in the form of diffusion equations with applications in modelling group dynamics and information flow in social networks as well as performing local and non-local image processing.