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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Brin, S., Page, L.: The anatomy of a large-scale hypertextual web search engine. Computer Networks 30(1-7), 101–117 (1998)
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)
Kannan, R., Vempala, S., Vetta, A.: On clusterings: Good, bad and spectral. Journal of the ACM 51(4), 540–556 (2004)
Kemeny, J.G., Snell, J.L.: Finite Markov Chains. D. VanNostrand Co, Inc., NewYork (1960)
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)
Chen, F., Lovász, L., Pak, I.: Lifting Markov chains to speed up mixing. In: Proceedings of STOC (1995)
Madras, N., Randall, D.: Markov Chain Decomposition for Convergence Rate Analysis. The Annals of Applied Probability 12(2), 581–606 (2002)
Mihail, M.: Conductance and Convergence of Markov Chains - A Combinatorial Treatment of Expanders. In: Proceedings of STOC (1989)
Montenegro, R.: Edge and vertex expansion bounds on eigenvalues of reversible Markov kernels (preprint)
Schweitzer, P.J.: Perturbation Theory and Finite Markov Chains. J. Applied Probability 5(3), 401–404 (1968)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ailon, N., Chien, S., Dwork, C. (2006). On Clusters in Markov Chains. In: Correa, J.R., Hevia, A., Kiwi, M. (eds) LATIN 2006: Theoretical Informatics. LATIN 2006. Lecture Notes in Computer Science, vol 3887. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11682462_9
Download citation
DOI: https://doi.org/10.1007/11682462_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-32755-4
Online ISBN: 978-3-540-32756-1
eBook Packages: Computer ScienceComputer Science (R0)