Generalizing Perron--Frobenius theory and eigenvector-based centralities to networks with complex edge weights

2026-06-10

AI Summary

This work addresses the limitation of classical Perron–Frobenius (PF) theory and eigenvector-based centrality measures, which are restricted to real-weighted networks and thus inapplicable to complex-weighted networks arising in quantum information, electrodynamics, and related fields. The study presents the first systematic formulation of a generalized PF theory for complex-valued matrices, establishing its connection to node importance in complex-weighted networks and introducing a corresponding generalized eigenvector centrality measure. Leveraging spectral theory for complex matrices, the authors prove the existence of complex-weighted networks satisfying generalized PF properties and demonstrate the efficacy of the proposed centrality metric through applications in electronic transport, circuit analysis, mathematical chemistry, and communication networks, thereby substantially extending the scope of classical network analysis methodologies.

Abstract

A fundamental concept in linear algebra and its applications to network analysis is the Perron--Frobenius (PF) theorem, which underpins eigenvector-based centrality measures such as eigenvector centrality, PageRank, and hubs and authorities. By invoking the PF theorem, we know for strongly connected networks with positive edge weights that the eigenvector corresponding to the largest eigenvalue of the weight matrix yields a well-defined centrality measure (namely, eigenvector centrality). Traditional formulations of the PF theorem and associated centrality measures assume that networks have real-valued weights. However, many networks in areas such as quantum information, quantum chemistry, electrodynamics, and machine learning have complex-valued edge weights. In this paper, we study generalizations of the PF theorem to complex-valued matrices, establish connections between these generalizations, and propose generalized eigenvector-based centrality measures to analyzing node importances in networks with complex edge weights. We also prove results about the existence of complex-weighted networks that satisfy generalized PF properties and calculate associated centrality measures for several examples, which we draw from application areas such as electron transport, circuit analysis, mathematical chemistry, and communication networks.

Problem

• Perron-Frobenius theory
• complex edge weights
• eigenvector centrality
• network analysis
• complex-valued matrices

Innovation

• Perron–Frobenius theorem
• complex-weighted networks
• eigenvector centrality
• network centrality
• complex matrices