On Kemeny's constant and stochastic complement

Given a stochastic matrix P partitioned in four blocks Pij, i,j=1,2, Kemeny's constant κ(P) is expressed in terms of Kemeny's constants of the stochastic complements P1=P11+P12(I−P22)−1P21, and P2=P22+P21(I−P11)−1P12. Specific cases concerning periodic Markov chains and Kronecker products...

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Bibliographic Details
Published in:Linear algebra and its applications Vol. 703; pp. 137 - 162
Main Authors: Bini, Dario Andrea, Durastante, Fabio, Kim, Sooyeong, Meini, Beatrice
Format: Journal Article
Language:English
Published: Elsevier Inc 15.12.2024
Subjects:
Divide-and-conquer algorithm
Kemeny's constant
Markov chains
65F15
Markov chains
65C40
Kemeny's constant
Divide-and-conquer algorithm
60J22
ISSN:0024-3795
Online Access:Get full text
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Summary:Given a stochastic matrix P partitioned in four blocks Pij, i,j=1,2, Kemeny's constant κ(P) is expressed in terms of Kemeny's constants of the stochastic complements P1=P11+P12(I−P22)−1P21, and P2=P22+P21(I−P11)−1P12. Specific cases concerning periodic Markov chains and Kronecker products of stochastic matrices are investigated. Bounds to Kemeny's constant of perturbed matrices are given. Relying on these theoretical results, a divide-and-conquer algorithm for the efficient computation of Kemeny's constant of graphs is designed. Numerical experiments performed on real world problems show the high efficiency and reliability of this algorithm. •Expression of Kemeny's constant employing the constants of stochastic complements.•New recursive algorithms for computing Kemeny's constant.•Application of the new expression to structured transition matrices.
ISSN:0024-3795
DOI:10.1016/j.laa.2024.09.001