Tensor Entropy for Uniform Hypergraphs
This work provides a novel entropy measure for uniform hypergraphs, addressing a gap in graph theory for higher-order structures, though it appears incremental as it builds on existing tensor methods.
The authors tackled the problem of defining entropy for uniform hypergraphs by extending von Neumann entropy using tensor theory, resulting in a measure that quantifies regularity and robustness with demonstrated bounds and efficient computation via tensor train decomposition.
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.