Dario Andrea Bini , Fabio Durastante , Sooyeong Kim , Beatrice Meini
{"title":"On Kemeny's constant and stochastic complement","authors":"Dario Andrea Bini , Fabio Durastante , Sooyeong Kim , Beatrice Meini","doi":"10.1016/j.laa.2024.09.001","DOIUrl":null,"url":null,"abstract":"<div><p>Given a stochastic matrix <em>P</em> partitioned in four blocks <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math></span>, <span><math><mi>i</mi><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></math></span>, Kemeny's constant <span><math><mi>κ</mi><mo>(</mo><mi>P</mi><mo>)</mo></math></span> is expressed in terms of Kemeny's constants of the stochastic complements <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>12</mn></mrow></msub><msup><mrow><mo>(</mo><mi>I</mi><mo>−</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>22</mn></mrow></msub><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>P</mi></mrow><mrow><mn>21</mn></mrow></msub></math></span>, and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>22</mn></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>21</mn></mrow></msub><msup><mrow><mo>(</mo><mi>I</mi><mo>−</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>P</mi></mrow><mrow><mn>12</mn></mrow></msub></math></span>. 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.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0024379524003574/pdfft?md5=c934bca31b23f0a0ae92728ad85a070b&pid=1-s2.0-S0024379524003574-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379524003574","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
Given a stochastic matrix P partitioned in four blocks , , Kemeny's constant is expressed in terms of Kemeny's constants of the stochastic complements , and . 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.
期刊介绍：
Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
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