Perturbation Analysis for Stationary Distributions of Markov Chains with Damping Component

Abstract

Perturbed Markov chains are popular models for description of information networks. In such models, the transition matrix \(\mathbf {P}_0\) of an information Markov chain is usually approximated by matrix \(\mathbf {P}_{\varepsilon } = (1 - \varepsilon ) \mathbf {P}_0 + \varepsilon \mathbf {D}\), where \(\mathbf {D}\) is a so-called damping stochastic matrix with identical rows and all positive elements, while \(\varepsilon \in [0, 1\)] is a damping (perturbation) parameter. We perform a detailed perturbation analysis for stationary distributions of such Markov chains, in particular get effective explicit series representations for the corresponding stationary distributions \(\bar{\pi }_\varepsilon \), upper bounds for the deviation \(| \bar{\pi }_{\varepsilon }- \bar{\pi }_0 |\), and asymptotic expansions for \(\bar{\pi }_{\varepsilon }\) with respect to the perturbation parameter \(\varepsilon \).