Hodge Decompositions for Weighted Hypergraphs

TL;DR

This paper extends Hodge decomposition theory from weighted simplicial complexes to weighted hypergraphs, exploring their Laplacians, homology, and eigenvalues to deepen understanding of hypergraph structures.

Contribution

It introduces a generalization of weighted Laplacians and Hodge decompositions from simplicial complexes to weighted hypergraphs, linking these operators with hypergraph homology.

Findings

Established relations between weighted Laplacians and embedded homology of hypergraphs.

Generalized Hodge decompositions to weighted hypergraphs.

Provided results on the eigenvalues of weighted hypergraph Laplacians.

Abstract

Weighted hypergraphs are generalizations of weighted simplicial complexes. In recent years, weighted Laplacians of weighted simplicial complexes have been studied. In 2016, as a generalization of the homology of simplicial complexes, the embedded homology of hypergraphs was constructed. In this paper, we generalize the weighted Laplacians of weighted simplicial complexes to weighted hypergraphs. We study the relations between the weighted Laplacians and the weighted embedded homology of weighted hypergraphs. We generalize the Hodge decompositions of weighted simplicial complexes to weighted hypergraphs. Moreover, as a complement for the Hodge decompositions, we give some results for the nonzero eigenvalues of the weighted Laplacians of weighted hypergraphs.