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Dirichlet Eigenvalues, Local Random Walks, and Analyzing Clusters in Graphs

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Algorithms and Computation (ISAAC 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8889))

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Abstract

A cluster \(S\) in a massive graph \(G\) is characterised by the property that its corresponding vertices are better connected with each other, in comparison with the other vertices of the graph. Modeling, finding and analyzing clusters in massive graphs is an important topic in various disciplines. In this work we study local random walks that always stay in a cluster \(S\). Moreover, we initiate the study of the local mixing time and the almost stable distribution, by analyzing Dirichlet eigenvalues in graphs. We prove that the Dirichlet eigenvalues of any connected subset \(S\) can be used to bound the \(\epsilon \)-uniform mixing time, which improves the previous best-known result. We further present two applications of our results. The first is a polynomial-time algorithm for finding clusters with an improved approximation guarantee, while the second is the significance ordering of vertices in a cluster.

This work has been partially funded by the Cluster of Excellence “Multimodal Computing and Interaction” within the Excellence Initiative of the German Federal Government.

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Authors and Affiliations

  1. Max Planck Institute für Informatics, 66123, Saarbrücken, Germany

    Pavel Kolev & He Sun

Authors
  1. Pavel Kolev
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  2. He Sun
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Correspondence to Pavel Kolev .

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Editors and Affiliations

  1. Pohang University of Science and Technology, Pohang, Korea, Republic of (South Korea)

    Hee-Kap Ahn

  2. Hankuk University of Foreign Studies, Yongin-si, Korea, Republic of (South Korea)

    Chan-Su Shin

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Kolev, P., Sun, H. (2014). Dirichlet Eigenvalues, Local Random Walks, and Analyzing Clusters in Graphs. In: Ahn, HK., Shin, CS. (eds) Algorithms and Computation. ISAAC 2014. Lecture Notes in Computer Science(), vol 8889. Springer, Cham. https://doi.org/10.1007/978-3-319-13075-0_49

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  • DOI: https://doi.org/10.1007/978-3-319-13075-0_49

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-13074-3

  • Online ISBN: 978-3-319-13075-0

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