  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
