Papers › On Kemeny's constant and stochastic complement
On Kemeny's constant and stochastic complement
Dario Andrea Bini, Fabio Durastante, Sooyeong Kim, Beatrice Meini
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Given a stochastic matrix P partitioned in four blocks Pᵢⱼ, i,j=1,2, Kemeny's constant κ(P) is expressed in terms of Kemeny's constants of the stochastic complements P₁=P₁₁+P₁₂(I-P₂₂)⁻¹P₂₁, and P₂=P₂₂+P₂₁(I-P₁₁)⁻¹P₁₂. 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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