Hypergraph $p$-Laplacians and Scale Spaces

TL;DR

This paper develops new mathematical tools for hypergraph analysis, including gradient and Laplacian operators, enabling advanced modeling of dynamics and information flow in complex networks and image processing.

Contribution

It introduces novel gradient, adjoint, and $p$-Laplacian definitions for both oriented and unoriented hypergraphs, expanding the analytical framework.

Findings

Defined hypergraph gradient and Laplacian operators.

Applied these operators to model diffusion and flow in networks.

Demonstrated applications in social network dynamics and image processing.

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.