Kronecker Product of Tensors and Hypergraphs: Structure and Dynamics
arxiv(2023)
摘要
Hypergraphs and tensors extend classic graph and matrix theory to account for
multiway relationships, which are ubiquitous in engineering, biological, and
social systems. While the Kronecker product is a potent tool for analyzing the
coupling of systems in graph or matrix contexts, its effectiveness in capturing
multiway interactions remains elusive. In this article, we present a
comprehensive exploration of algebraic, structural, and spectral properties of
the tensor Kronecker product. We express Tucker and tensor train decompositions
and various tensor eigenvalues in terms of the tensor Kronecker product.
Additionally, we utilize the tensor Kronecker product to form Kronecker
hypergraphs, a tensor-based hypergraph product, and investigate the structure
and stability of polynomial dynamics on Kronecker hypergraphs. Finally, we
provide numerical examples to demonstrate the utility of the tensor Kronecker
product in computing Z-eigenvectors, various tensor decompositions, and
determining the stability of polynomial systems.
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