Tensor Entropy for Uniform Hypergraphs

TL;DR

This paper introduces a tensor-based entropy measure for uniform hypergraphs, extending von Neumann entropy, and demonstrates its effectiveness in quantifying regularity and robustness through theoretical bounds and experiments.

Contribution

It develops a novel tensor entropy for hypergraphs using tensor SVD, extending graph entropy concepts and providing efficient computation methods.

Findings

Tensor entropy extends von Neumann entropy to hypergraphs.

Entropy bounds relate to hypergraph regularity.

Tensor train decomposition enables efficient computation.

Abstract

In this paper, we develop the notion of entropy for uniform hypergraphs via tensor theory. We employ the probability distribution of the generalized singular values, calculated from the higher-order singular value decomposition of the Laplacian tensors, to fit into the Shannon entropy formula. We show that this tensor entropy is an extension of von Neumann entropy for graphs. In addition, we establish results on the lower and upper bounds of the entropy and demonstrate that it is a measure of regularity for uniform hypergraphs in simulated and experimental data. We exploit the tensor train decomposition in computing the proposed tensor entropy efficiently. Finally, we introduce the notion of robustness for uniform hypergraphs.