On Kemeny's constant and stochastic complement

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.