Implications

  • Behrouz Touri
Chapter
Part of the Springer Theses book series (Springer Theses)

Abstract

In this chapter, we discuss some of the implications of the results developed in Chaps. 3 and 4. We first study the implications of the developed results on the product of independent random stochastic chains in Sect. 5.1, and also develop a rate of convergence result for ergodic independent random chains. Then in Sect. 5.2, we study the implication of Theorem 4.10 in non-negative matrix theory. There, we show that Theorem 4.10 is an extension of a well-known result for homogeneous Markov chains to inhomogeneous products of stochastic matrices. In Sect. 5.3, we provide a convergence rate analysis for averaging dynamics driven by uniformly bounded chains. Then, in Sect. 5.4, we introduce link-failure models for random chains and analyze the effect of link-failure on the limiting behavior of averaging dynamics. In Sect. 5.5, we study the Hegselmann-Krause model for opinion dynamics in social networks and provide a new bound on the termination time of such dynamics. Finally, in Sect. 5.6, using the developed tools, we propose an alternative proof for the second Borel-Cantelli lemma.

Keywords

Convergence Result Termination Time Final Component Trivial Event Stochastic Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Behrouz Touri
    • 1
  1. 1.University of Illinois at Urbana-ChampaignUrbanaUSA

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