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On Clusters in Markov Chains

  • Nir Ailon
  • Steve Chien
  • Cynthia Dwork
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3887)

Abstract

Motivated by the computational difficulty of analyzing very large Markov chains, we define a notion of clusters in (not necessarily reversible) Markov chains, and explore the possibility of analyzing a cluster “in vitro,” without regard to the remainder of the chain. We estimate the stationary probabilities of the states in the cluster using only transition information for these states, and bound the error of the estimate in terms of parameters measuring the quality of the cluster. Finally, we relate our results to searching in a hyperlinked environment, and provide supporting experimental results.

Keywords

Markov Chain Stationary Distribution Fundamental Matrix Random Graph Model Finite Markov Chain 
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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References

  1. 1.
    Brin, S., Page, L.: The anatomy of a large-scale hypertextual web search engine. Computer Networks 30(1-7), 101–117 (1998)Google Scholar
  2. 2.
    Flake, G., Lawrence, S., Giles, C.: Efficient Identification of Web Communities. In: Proceedings of the Sixth International Conference on Knowledge Discovery and Data Mining (ACM SIGKDD 2000), pp. 150–160 (2000)Google Scholar
  3. 3.
    Kannan, R., Vempala, S., Vetta, A.: On clusterings: Good, bad and spectral. Journal of the ACM 51(4), 540–556 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kemeny, J.G., Snell, J.L.: Finite Markov Chains. D. VanNostrand Co, Inc., NewYork (1960)zbMATHGoogle Scholar
  5. 5.
    Lovász, L., Winkler, P.: “Mixing times”, Microsurveys in Discrete Probability. In: Aldous, D., Propp, J. (eds.) DIMACS Series in Discrete Mathematics and Theoretical Computer Science, AMS, pp. 85–133 (1998)Google Scholar
  6. 6.
    Chen, F., Lovász, L., Pak, I.: Lifting Markov chains to speed up mixing. In: Proceedings of STOC (1995)Google Scholar
  7. 7.
    Madras, N., Randall, D.: Markov Chain Decomposition for Convergence Rate Analysis. The Annals of Applied Probability 12(2), 581–606 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mihail, M.: Conductance and Convergence of Markov Chains - A Combinatorial Treatment of Expanders. In: Proceedings of STOC (1989)Google Scholar
  9. 9.
    Montenegro, R.: Edge and vertex expansion bounds on eigenvalues of reversible Markov kernels (preprint)Google Scholar
  10. 10.
    Schweitzer, P.J.: Perturbation Theory and Finite Markov Chains. J. Applied Probability 5(3), 401–404 (1968)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Nir Ailon
    • 1
  • Steve Chien
    • 2
  • Cynthia Dwork
    • 2
  1. 1.Princeton UniversityPrincetonUSA
  2. 2.Microsoft ResearchMountain ViewUSA

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