On Kemeny's constant and stochastic complement

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2024-09-05 DOI:10.1016/j.laa.2024.09.001

Dario Andrea Bini , Fabio Durastante , Sooyeong Kim , Beatrice Meini

引用次数: 0

Abstract

Given a stochastic matrix P partitioned in four blocks $P_{i j}$ , $i, j = 1, 2$ , Kemeny's constant $κ (P)$ is expressed in terms of Kemeny's constants of the stochastic complements $P_{1} = P_{11} + P_{12} {(I - P_{22})}^{- 1} P_{21}$ , and $P_{2} = P_{22} + P_{21} {(I - P_{11})}^{- 1} P_{12}$ . 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.

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关于凯梅尼常数和随机补数

鉴于随机矩阵 P 分成四个块 Pij（i,j=1,2），可门尼常数 κ(P)用随机补集 P1=P11+P12(I-P22)-1P21 和 P2=P22+P21(I-P11)-1P12 的可门尼常数表示。研究了周期马尔可夫链和随机矩阵的克朗克积的具体情况。给出了扰动矩阵的凯美尼常数的界值。根据这些理论结果，设计了一种高效计算图的凯门尼常数的分而治之算法。在实际问题上进行的数值实验表明了该算法的高效性和可靠性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

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.