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Perturbation Analysis for Stationary Distributions of Markov Chains with Damping Component

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Algebraic Structures and Applications (SPAS 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 317))

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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 \(\varepsilon \).

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Acknowledgements

This research was supported by the Swedish International Development Cooperation Agency (Sida), International Science Programme (ISP) in Mathematical Sciences (IPMS) and Sida Bilateral Research Programmes for research and education capacity development in Mathematics in Uganda and Tanzania. The authors are also grateful to the research environment Mathematics and Applied Mathematics (MAM), Division of Applied Mathematics, Mälardalen University for providing an excellent and inspiring environment for research education and research.

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Authors and Affiliations

  1. Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden

    Dmitrii Silvestrov

  2. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Box 883, 72123, Västerås, Sweden

    Sergei Silvestrov, Benard Abola, Pitos Seleka Biganda & Christopher Engström

  3. Department of Mathematics, School of Physical Sciences, Makerere University, Box 7062, Kampala, Uganda

    John Magero Mango & Godwin Kakuba

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  1. Dmitrii Silvestrov
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Correspondence to Dmitrii Silvestrov .

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Editors and Affiliations

  1. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Sergei Silvestrov

  2. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Anatoliy Malyarenko

  3. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Milica Rančić

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Silvestrov, D. et al. (2020). Perturbation Analysis for Stationary Distributions of Markov Chains with Damping Component. In: Silvestrov, S., Malyarenko, A., Rančić, M. (eds) Algebraic Structures and Applications. SPAS 2017. Springer Proceedings in Mathematics & Statistics, vol 317. Springer, Cham. https://doi.org/10.1007/978-3-030-41850-2_38

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