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arxiv: 2606.12026 · v1 · pith:6LQU4CI6new · submitted 2026-06-10 · 🧮 math.SP · cs.SI· math-ph· math.MP· physics.data-an

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

Yu Tian , Mason A. Porter , Lucas B\"ottcher This is my paper

Pith reviewed 2026-06-27 07:43 UTC · model grok-4.3

classification 🧮 math.SP cs.SImath-phmath.MPphysics.data-an
keywords Perron-Frobenius theoremcomplex edge weightseigenvector centralityweighted networksspectral graph theorynetwork centrality
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The pith

Generalizations of the Perron-Frobenius theorem enable eigenvector centrality measures for networks whose edges carry complex weights.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends classical Perron-Frobenius results from real nonnegative matrices to complex-valued matrices. It links several such generalizations and shows how their associated eigenvectors can serve as centrality scores that rank node importance in networks with complex edge weights. The authors prove that networks satisfying the required connectivity conditions exist in the complex setting and compute the resulting centralities on examples drawn from electron transport, circuit analysis, mathematical chemistry, and communication networks.

Core claim

Generalizations of the Perron-Frobenius theorem to irreducible complex matrices guarantee a distinguished eigenvalue whose eigenvector yields a well-defined centrality measure; the paper establishes connections among these generalizations, proves existence of complex-weighted networks that meet the necessary conditions, and applies the measures to concrete examples.

What carries the argument

Generalized Perron-Frobenius theorems for complex matrices, which identify a spectral radius eigenvalue whose eigenvector supplies node centrality scores.

If this is right

  • Eigenvector centrality becomes applicable to networks in quantum information and electrodynamics.
  • Node importances in electron transport and circuit networks can be ranked using the generalized eigenvectors.
  • Existence proofs confirm that the required conditions hold for some realistic complex-weighted systems.
  • Centrality calculations are feasible for communication networks with complex edge weights.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The complex-valued scores may encode phase or interference information absent from real-weighted centralities.
  • The framework could be tested on non-Hermitian operators arising in open quantum systems.
  • Similar generalizations might apply to adjacency tensors in higher-order networks with complex weights.

Load-bearing premise

The complex-valued weight matrices must satisfy the irreducibility or strong-connectivity conditions required by the chosen generalizations of the Perron-Frobenius theorem.

What would settle it

A strongly connected complex-weighted network whose adjacency matrix fails to possess an eigenvector corresponding to the spectral radius that produces consistent centrality rankings under any of the studied generalizations.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript generalizes the Perron-Frobenius theorem to complex-valued matrices, establishes connections among these generalizations, proposes generalized eigenvector-based centrality measures for networks with complex edge weights, proves existence results for complex-weighted networks satisfying the generalized PF properties, and computes the associated centrality measures for several examples drawn from electron transport, circuit analysis, mathematical chemistry, and communication networks.

Significance. If the central claims hold, the work extends eigenvector centrality to complex-weighted networks arising in quantum information, chemistry, and related fields, providing both theoretical connections and concrete existence proofs plus example calculations. The absence of free parameters or ad-hoc axioms in the core theory is a strength.

major comments (1)
  1. [Examples section] Examples section (referenced in the abstract as the calculations for several examples): the paper invokes generalized PF theorems (e.g., eventual positivity or spectral-radius dominance) to define and compute centrality measures, yet provides no explicit verification that any of the example weight matrices are irreducible or that the underlying digraphs are strongly connected. These hypotheses are required for the uniqueness and positivity properties that underpin the centrality interpretation; without them the reported eigenvectors are not guaranteed to satisfy the claimed properties.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for providing constructive comments. We respond to the major comment as follows.

read point-by-point responses

  1. Referee: [Examples section] Examples section (referenced in the abstract as the calculations for several examples): the paper invokes generalized PF theorems (e.g., eventual positivity or spectral-radius dominance) to define and compute centrality measures, yet provides no explicit verification that any of the example weight matrices are irreducible or that the underlying digraphs are strongly connected. These hypotheses are required for the uniqueness and positivity properties that underpin the centrality interpretation; without them the reported eigenvectors are not guaranteed to satisfy the claimed properties.

    Authors: We acknowledge the referee's observation. The examples were chosen from application domains where the underlying networks satisfy the necessary conditions for the generalized Perron-Frobenius theorems to apply, including irreducibility. However, we agree that explicit verification should be provided to make this clear to readers. In the revised manuscript, we will include a brief verification or statement for each example confirming that the weight matrices are irreducible and the digraphs are strongly connected. This will support the claimed properties of the centrality measures. revision: yes

Circularity Check

0 steps flagged

No significant circularity; theoretical generalization is self-contained

full rationale

The paper generalizes the standard Perron-Frobenius theorem to complex matrices, establishes connections between generalizations, proposes centrality measures, proves existence of networks satisfying the properties, and computes examples. No fitted parameters, self-definitional constructs, or load-bearing self-citations appear in the provided text. Derivations rely on external mathematical results without reducing claims to tautologies by construction. The skeptic concern addresses applicability of irreducibility conditions to examples but does not indicate circularity in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the paper relies on standard linear-algebra axioms for complex matrices and existing generalizations of the Perron-Frobenius theorem; no free parameters, ad-hoc axioms, or invented entities are mentioned.

axioms (1)
  • standard math Standard properties of eigenvalues and eigenvectors for complex matrices from linear algebra
    The generalizations invoked in the abstract rest on classical results about complex matrices.

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discussion (0)

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Reference graph

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