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Perturbed Markov chains are popular models for description of information networks. In such models, the transition matrix P0 of an information Markov chain is usually approximated by matrix P_ε = (1 − ε)P₀ + εD, where D is a so-called damping stochastic matrix with identical rows and all positive elements, while ε ∈ [0, 1] is a damping (perturbation) parameter. Using procedure of artificial regeneration for the perturbed Markov chain ηε,n, with the matrix of transition probabilities P_ε, and coupling methods, we get ergodic theorems, in the form of asymptotic relations for p_ε,ij (n) = P_i {η_ε,n = j} as n → ∞ and ε → 0, and explicit upper bounds for the rates of convergence in such theorems. In particular, the most difficult case of the model with singular perturbations, where the phase space of the unperturbed Markov chain η₀,n split in several closed classes of communicative states and possibly a class of transient states, is investigated.