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A Perron-Frobenius Theorem for Strongly Aperiodic Stochastic Chains

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Rohit Parasnis, Massimo Franceschetti, Behrouz Touri

Optimization and Control · Submitted 1 Apr 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We derive a generalization of the Perron-Frobenius theorem to time-varying row-stochastic matrices as follows: using Kolmogorov's concept of absolute probability sequences, which are time-varying analogs of principal eigenvectors, we identify a set of connectivity conditions that generalize the notion of irreducibility (strong connectivity) to time-varying matrices (networks), and we show that under these conditions, the absolute probability sequence associated with a given matrix sequence is (a) uniformly positive and (b) unique.

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arXiv ID2204.00573 (version 2)
Pith IDZHOXMA6F, Pith's permanent record of this paper and its review
CitationBehrouz Touri, Massimo Franceschetti, Rohit Parasnis. "A Perron-Frobenius Theorem for Strongly Aperiodic Stochastic Chains." arXiv:2204.00573 (2022). https://pith.science/paper/ZHOXMA6F
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arXiv listing abstract v2. Results and the main claim are underlined; .

We derive a generalization of the Perron-Frobenius theorem to time-varying row-stochastic matrices as follows: using Kolmogorov's concept of absolute probability sequences, which are time-varying analogs of principal eigenvectors, we identify a set of connectivity conditions that generalize the notion of irreducibility (strong connectivity) to time-varying matrices (networks), and we show that under these conditions, the absolute probability sequence associated with a given matrix sequence is (a) uniformly positive and (b) unique. Our results apply to both discrete-time and continuous-time settings. We then discuss a few applications of our main results to non-Bayesian learning, distributed optimization, opinion dynamics, and averaging dynamics over random networks.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We derive a generalization of the Perron-Frobenius theorem to time-varying row-stochastic matrices as follows: using Kolmogorov's concept of absolute probability sequences, which are time-varying analogs of principal eigenvectors, we identify a set of connectivity conditions that generalize the notion of irreducibility (strong connectivity) to time-varying matrices (networks), and we show that under these conditions, the absolute probability sequence associated with a given matrix sequence is (a) uniformly positive and (b) unique. contribution 0.90 0.96 0.91 asserted 1.00
s1 Our results apply to both discrete-time and continuous-time settings. result 0.41 0.74 0.04 asserted 1.00
s2 We then discuss a few applications of our main results to non-Bayesian learning, distributed optimization, opinion dynamics, and averaging dynamics over random networks. contribution 0.87 0.82 0.05 asserted 0.92

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Original abstract

We derive a generalization of the Perron-Frobenius theorem to time-varying row-stochastic matrices as follows: using Kolmogorov's concept of absolute probability sequences, which are time-varying analogs of principal eigenvectors, we identify a set of connectivity conditions that generalize the notion of irreducibility (strong connectivity) to time-varying matrices (networks), and we show that under these conditions, the absolute probability sequence associated with a given matrix sequence is (a) uniformly positive and (b) unique. Our results apply to both discrete-time and continuous-time settings. We then discuss a few applications of our main results to non-Bayesian learning, distributed optimization, opinion dynamics, and averaging dynamics over random networks.

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