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Perturbation Analysis for Stationary Distributions of Markov Chains with Damping Component

  • Dmitrii SilvestrovEmail author
  • Sergei Silvestrov
  • Benard Abola
  • Pitos Seleka Biganda
  • Christopher Engström
  • John Magero Mango
  • Godwin Kakuba
Conference paper
  • 45 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 317)

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 \(\varepsilon \).

Keywords

Markov chain Damping component Information network Regular perturbation Singular perturbation Stationary distribution Rate of convergence Asymptotic expansion 

MSC 2010 Classification

60J10 

Notes

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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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Dmitrii Silvestrov
    • 1
    Email author
  • Sergei Silvestrov
    • 2
  • Benard Abola
    • 2
  • Pitos Seleka Biganda
    • 2
  • Christopher Engström
    • 2
  • John Magero Mango
    • 3
  • Godwin Kakuba
    • 3
  1. 1.Department of MathematicsStockholm UniversityStockholmSweden
  2. 2.Division of Applied Mathematics, School of Education, Culture and CommunicationMälardalen UniversityVästeråsSweden
  3. 3.Department of Mathematics, School of Physical SciencesMakerere UniversityKampalaUganda

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