Researchers Extend Perron-Frobenius Theory to Networks with Complex Edge Weights
A new preprint generalizes the Perron–Frobenius theorem and eigenvector-based centrality measures to networks with complex-valued edge weights. The Perron–Frobenius theorem has long underpinned tools like PageRank and eigenvector centrality, but was previously limited to real-valued network weights. The work opens the door to rigorous node-importance analysis in quantum, electrical, and machine-learning networks where edge weights are inherently complex numbers.
Researchers have posted a 34-page preprint on arXiv extending the classical Perron–Frobenius (PF) theorem—a cornerstone of linear algebra used to justify eigenvector centrality, PageRank, and hubs-and-authorities measures—to matrices with complex-valued entries. Traditional formulations of the theorem require real, typically positive, edge weights, which excludes many physically and computationally important systems. The paper establishes several generalizations of the PF theorem for complex-valued matrices, proves connections among them, and proposes corresponding generalized eigenvector-based centrality measures for quantifying node importance in such networks. The authors also prove existence results for complex-weighted networks satisfying these generalized PF properties and demonstrate the framework on examples drawn from electron transport, circuit analysis, mathematical chemistry, and communication networks. The work addresses a growing need in fields such as quantum information, quantum chemistry, and electrodynamics, where network representations naturally carry complex weights.
What's missing
As a preprint, the paper has not yet undergone formal peer review, so the correctness and completeness of the proofs have not been independently verified. The authors do not appear to discuss computational complexity or scalability of the proposed centrality measures for large real-world networks, nor do they benchmark against existing heuristic approaches for complex-weighted graphs.
What different sources said
- arXiv physicsCenter
Generalizing Perron--Frobenius theory and eigenvector-based centralities to networks with complex edge weights
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