Perturbed Markov Chains and Information Networks

TL;DR

This paper analyzes how small perturbations in Markov chains, used for modeling information networks, affect their stationary distributions and convergence rates, providing bounds, asymptotic expansions, and numerical validation.

Contribution

It introduces effective bounds and asymptotic formulas for stationary distributions of perturbed Markov chains, enhancing understanding of their convergence behavior.

Findings

Effective upper bounds for stationary distribution approximation.

Asymptotic expansions with respect to damping parameter.

Numerical experiments validating theoretical results.

Abstract

The paper is devoted to studies of perturbed Markov chains commonly used for description of information networks. In such models, the matrix of transition probabilities for the corresponding Markov chain is usually regularised by adding a special damping matrix multiplied by a small damping (perturbation) parameter $\varepsilon$. We give effective upper bounds for the rate of approximation for stationary distributions of unperturbed Markov chains by stationary distributions of perturbed Markov chains with regularised matrices of transition probabilities, asymptotic expansions for approximating stationary distributions with respect to damping parameter, as well as explicit upper bounds for the rate of convergence in ergodic theorems for $n$-step transition probabilities in triangular array mode, where perturbation parameter $\varepsilon \to 0$ and $n \to \infty$, simultaneously. The results of numerical experiments are also presented