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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Veröffentlicht in:	Linear algebra and its applications Jg. 703; S. 137 - 162
Hauptverfasser:	Bini, Dario Andrea, Durastante, Fabio, Kim, Sooyeong, Meini, Beatrice
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Elsevier Inc 15.12.2024
Schlagworte:	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-Zugang:	Volltext
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Zusammenfassung:	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