A Perron-Frobenius Theorem for Strongly Aperiodic Stochastic Chains

• Published in: IEEE transactions on automatic control Vol. 70; no. 7; pp. 4286 - 4301
• Main Authors: Parasnis, Rohit, Franceschetti, Massimo, Touri, Behrouz
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.07.2025; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Consensus algorithms; Convergence; decentralized control; distributed algorithms; Dynamical systems; Eigenvectors; Heuristic algorithms; Indexes; Linear matrix inequalities; networked control systems; Optimization; Stochastic processes; Sufficient conditions; Theorems; Time-varying systems; Vectors
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2025.3527332

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.