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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| Vydáno v: | Linear algebra and its applications Ročník 703; s. 137 - 162 |
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15.12.2024
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Bini, Dario Andrea Kim, Sooyeong Meini, Beatrice Durastante, Fabio |
| Author_xml | – sequence: 1 givenname: Dario Andrea surname: Bini fullname: Bini, Dario Andrea email: dario.bini@unipi.it organization: Mathematics Department, University of Pisa, Largo Bruno Pontecorvo, 5, Pisa, 56127, PI, Italy – sequence: 2 givenname: Fabio orcidid: 0000-0002-1412-8289 surname: Durastante fullname: Durastante, Fabio email: fabio.durastante@unipi.it organization: Mathematics Department, University of Pisa, Largo Bruno Pontecorvo, 5, Pisa, 56127, PI, Italy – sequence: 3 givenname: Sooyeong surname: Kim fullname: Kim, Sooyeong email: kimswim@yorku.ca organization: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, M3J 1P3, ON, Canada – sequence: 4 givenname: Beatrice surname: Meini fullname: Meini, Beatrice email: beatrice.meini@unipi.it organization: Mathematics Department, University of Pisa, Largo Bruno Pontecorvo, 5, Pisa, 56127, PI, Italy |
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| Keywords | 65F15 Markov chains 65C40 Kemeny's constant Divide-and-conquer algorithm 60J22 |
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| Snippet | 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... |
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| SubjectTerms | Divide-and-conquer algorithm Kemeny's constant Markov chains |
| Title | On Kemeny's constant and stochastic complement |
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