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(1)1+ P-12(I-P-22)-1P(21), and P-2 = P-22+P-21(I-P-11)-1P(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. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).