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On Kemeny's constant and stochastic complement

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

TL;DR

This work studies the computation of Kemeny’s constant $\kappa(P)$ for irreducible finite Markov chains by expressing it in terms of the stochastic complements $P_1$ and $P_2$ of a block partition of $P$, plus a correction term $\gamma$. The authors derive a general decomposition $\kappa(P) = \kappa(P_1) + \kappa(P_2) + \gamma$ and provide explicit forms for $\gamma$, along with closed-form results for structured cases including periodic chains, Kronecker products, and sub-stochastic blocks with constant row sums. Building on this, they propose a divide-and-conquer algorithm that recursively computes Kemeny’s constant by solving sparse linear systems on smaller blocks and combining results via $\gamma$, and they augment this with a low-precision randomized trace estimator to accelerate large-scale computation. Numerical experiments on real-world graphs demonstrate substantial speedups and reliability, validating the approach for fast connectivity-based metrics in graphs and related applications.

Abstract

Given a stochastic matrix $P$ partitioned in four blocks $P_{ij}$, $i,j=1,2$, 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.

On Kemeny's constant and stochastic complement

TL;DR

This work studies the computation of Kemeny’s constant for irreducible finite Markov chains by expressing it in terms of the stochastic complements and of a block partition of , plus a correction term . The authors derive a general decomposition and provide explicit forms for , along with closed-form results for structured cases including periodic chains, Kronecker products, and sub-stochastic blocks with constant row sums. Building on this, they propose a divide-and-conquer algorithm that recursively computes Kemeny’s constant by solving sparse linear systems on smaller blocks and combining results via , and they augment this with a low-precision randomized trace estimator to accelerate large-scale computation. Numerical experiments on real-world graphs demonstrate substantial speedups and reliability, validating the approach for fast connectivity-based metrics in graphs and related applications.

Abstract

Given a stochastic matrix partitioned in four blocks , , Kemeny's constant is expressed in terms of Kemeny's constants of the stochastic complements , and . 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.
Paper Structure (18 sections, 12 theorems, 94 equations, 1 figure, 2 tables, 1 algorithm)

This paper contains 18 sections, 12 theorems, 94 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Stochastic complement
  4. Kemeny's constant
  5. Kemeny's constant and graphs
  6. Kemeny's constant and stochastic complement
  7. Application of Theorem \ref{['th:kappap']}
  8. Periodic Markov chains
  9. Kronecker product of stochastic matrices
  10. Sub-stochastic matrices with constant row sums
  11. Some bounds
  12. A perturbation result
  13. A divide-and-conquer algorithm
  14. Low-precision randomized approximation
  15. Numerical examples
  16. ...and 3 more sections

Key Result

Lemma 2.1

Let $\mathbf{g},\mathbf{h}\in\mathbb{R}^{n}$ be vectors with $\mathbf{h}^T\mathbf{g}=1$, $\mathbf{h}^T\mathop{\mathrm{\mathbf{1}}}\nolimits \neq 0$, $\boldsymbol{\pi}^T \mathbf{g} \neq 0$. Then, $I-P+\mathbf{g}\mathbf{h}^T$ is non-singular, and where $Z = (I-P+\mathbf{g}\mathbf{h}^T)^{-1}$.

Figures (1)

  • Figure 1: Application of the Nested Dissection algorithm from MR1639073 to the Gaertner/nopoly matrix from the SuiteSparse Matrix Collection (formerly the University of Florida Sparse Matrix Collection) MR2865011.

Theorems & Definitions (24)

  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 4.1
  • ...and 14 more