Hypergraph Analysis Based on a Compatible Tensor Product Structure

TL;DR

This paper introduces a novel tensor product structure for hypergraphs, linking algebraic connectivity with vertex connectivity, and applies it to optimize hypergraph connectivity and embedding algorithms.

Contribution

It presents a new compatible tensor product for hypergraphs, defines algebraic connectivity in this context, and integrates it into connectivity optimization and Laplacian eigenmap algorithms.

Findings

Established relationship between algebraic and vertex connectivity in hypergraphs.

Developed methods for hypergraph connectivity optimization using algebraic connectivity.

Extended Laplacian eigenmap algorithm to hypergraphs with the new tensor product.

Abstract

We propose a tensor product structure that is compatible with the hypergraph structure. We define the algebraic connectivity of the $(m+1)$-uniform hypergraph in this product, and prove the relationship with the vertex connectivity. We introduce some connectivity optimization problem into the hypergraph, and solve them with the algebraic connectivity. We introduce the Laplacian eigenmap algorithm to the hypergraph under our tensor product.