Given a stochastic matrix $P$ partitioned in four blocks $P_{ij}$, $i,j=1,2$, Kemeny's constant $\kappa(P)$ 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.

翻译：给定一个分块为四个子矩阵$P_{ij}$（$i,j=1,2$）的随机矩阵$P$，Kemeny常数$\kappa(P)$可用随机补$P_1=P_{11}+P_{12}(I-P_{22})^{-1}P_{21}$和$P_2=P_{22}+P_{21}(I-P_{11})^{-1}P_{12}$的Kemeny常数表示。针对周期马尔可夫链及随机矩阵的Kronecker积等特例进行了研究，并给出了扰动矩阵Kemeny常数的界。基于这些理论结果，设计了一种用于高效计算图Kemeny常数的分治算法。实际问题的数值实验表明该算法具有高效性与可靠性。