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Perron-Frobenius theorem for dual tensors and its applications

Changjiang Bu, Yue Chu, Qingying Zhang, Jiang Zhou

TL;DR

This work extends the Perron-Frobenius theory to dual tensors by formulating a dual-number tensor framework and linking spectral properties to hypergraph centrality. It proves the existence of a positive dual eigenpair for $\mathcal{A}=\mathcal{A}_s+\mathcal{A}_d\epsilon$ with $\lambda=\lambda_s+\lambda_d\epsilon$ and provides an explicit expression for the dual part via an $M$-matrix generalized inverse. The results yield a dual centrality concept for graphs and hypergraphs, with a concrete four-step computation that combines $\lambda_d$ and $x_d$ to form the centrality vector $x= x_s+x_d\epsilon$; perturbations can then distinguish vertices that are tied under $x_s$. Numerical experiments on regular graphs and hypergraphs demonstrate that dual perturbations reveal refined vertex rankings, highlighting the practical impact for network analysis.

Abstract

The Perron-Frobenius theorem of nonnegative matrices is a classical result on spectral theory of matrices, which has wide applications in many domains. In this paper, we give the Perron-Frobenius theorem for dual tensors, that is, a dual tensor with weakly irreducible nonnegative standard part has a positive dual eigenvalue with a positive dual eigenvector. We give an explicit formula for the dual part of the positive dual eigenvector by using generalized inverses of an $M$-matrix. By considering the natural correspondence between tensors (matrices) and hypergraphs (graphs), some basic properties on the positive dual eigenvalue and positive dual eigenvector of hypergraphs are obtained. As applications, we introduce dual centrality measures for vertices of graphs and hypergraphs. By introducing a dual perturbation, vertices that are tied under eigenvector centrality can be effectively distinguished. In our numerical experiments, by perturbing specific structures, we successfully differentiated vertices in regular graphs and hypergraphs that were previously indistinguishable.

Perron-Frobenius theorem for dual tensors and its applications

TL;DR

This work extends the Perron-Frobenius theory to dual tensors by formulating a dual-number tensor framework and linking spectral properties to hypergraph centrality. It proves the existence of a positive dual eigenpair for with and provides an explicit expression for the dual part via an -matrix generalized inverse. The results yield a dual centrality concept for graphs and hypergraphs, with a concrete four-step computation that combines and to form the centrality vector ; perturbations can then distinguish vertices that are tied under . Numerical experiments on regular graphs and hypergraphs demonstrate that dual perturbations reveal refined vertex rankings, highlighting the practical impact for network analysis.

Abstract

The Perron-Frobenius theorem of nonnegative matrices is a classical result on spectral theory of matrices, which has wide applications in many domains. In this paper, we give the Perron-Frobenius theorem for dual tensors, that is, a dual tensor with weakly irreducible nonnegative standard part has a positive dual eigenvalue with a positive dual eigenvector. We give an explicit formula for the dual part of the positive dual eigenvector by using generalized inverses of an -matrix. By considering the natural correspondence between tensors (matrices) and hypergraphs (graphs), some basic properties on the positive dual eigenvalue and positive dual eigenvector of hypergraphs are obtained. As applications, we introduce dual centrality measures for vertices of graphs and hypergraphs. By introducing a dual perturbation, vertices that are tied under eigenvector centrality can be effectively distinguished. In our numerical experiments, by perturbing specific structures, we successfully differentiated vertices in regular graphs and hypergraphs that were previously indistinguishable.

Paper Structure

This paper contains 5 sections, 9 theorems, 30 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Eigenvalues of dual tensors
  4. Dual eigenvalues and eigenvectors of hypergraphs
  5. Dual Centrality of vertices in graphs and hypergraphs

Key Result

Lemma 2.1

Let $M$ be a real square matrix such that $\hbox{\rm rank}(M)=\hbox{\rm rank}(M^2)$. If $x\in R(M)$ and $y\in R(M^\top)$, then for any $\{1\}$-inverse $M^{(1)}$ of $M$.

Figures (2)

  • Figure 1: A regular garph with $8$ vertices.
  • Figure 2: A regular hypergarph with $9$ vertices.

Theorems & Definitions (13)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • ...and 3 more