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.
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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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References
Andersen, R., Chung, F.R.K., Lang, K.J.: Using pagerank to locally partition a graph. Internet Mathematics 4(1), 35–64 (2007)
Andersen, R., Peres, Y.: Finding sparse cuts locally using evolving sets. In: 41st Annual Symposium on Theory of Computing, STOC 2009, pp. 235–244 (2009)
Berenbrink, P., Cooper, C., Elsässer, R., Radzik, T., Sauerwald, T.: Speeding up random walks with neighborhood exploration. In: 21st Annual Symposium on Discrete Algorithms, SODA 2010, pp. 1422–1435 (2010)
Chung, F.R.K.: Spectral Graph Theory. American Mathematical Society (1997)
Chung, F.: Random walks and local cuts in graphs. Linear Algebra and its Applications 423(1), 22–32 (2007)
Chung, F., Yau, S.-T.: Discrete green’s functions. J. Comb. Theory Series A 91(1–2), 214 (2000)
Cooper, C., Elsässer, R., Ono, H., Radzik, T.: Coalescing random walks and voting on connected graphs. SIAM J. Discrete Math. 27(4), 1748–1758 (2013)
Elsässer, R., Sauerwald, T.: Tight bounds for the cover time of multiple random walks. Theor. Comput. Sci. 412(24), 2623–2641 (2011)
Friedrich, T., Sauerwald, T.: The cover time of deterministic random walks. Electr. J. Comb. 17(1) (2010)
Gharan, S.O., Trevisan, L.: Approximating the expansion profile and almost optimal local graph clustering. In: 53rd Annual Symposium on Foundations of Computer Science, FOCS 2012, pp. 187–196 (2012)
Jerrum, M., Sinclair, A.: Approximating the permanent. SIAM J. Comput. 18(6), 1149–1178 (1989)
Gharan, S.O., Trevisan, L.: Approximating the expansion profile and almost optimal local graph clustering. In: 53rd Annual Symposium on Foundations of Computer Science, FOCS 2012, pp. 187–196 (2012)
Lovász, L., Simonovits, M.: The mixing rate of markov chains, an isoperimetric inequality, and computing the volume. In: 31st Annual Symposium on Foundations of Computer Science, FOCS 1990, pp. 346–354 (1990)
Lovász, L., Simonovits, M.: Random walks in a convex body and an improved volume algorithm. Rand. Struct. & Algo. 4(4), 359–412 (1993)
Sauerwald, T., Sun, H.: Tight bounds for randomized load balancing on arbitrary network topologies. In: 53rd Annual Symposium on Foundations of Computer Science, FOCS 2012, pp. 341–350 (2012)
Spielman, D.A., Teng, S.-H.: Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In: 36th Annual Symposium on Theory of Computing, STOC 2004, pp. 81–90 (2004)
Zhu, Z.A., Lattanzi, S., Mirrokni, V.S.: A local algorithm for finding well-connected clusters. In: 30th International Conference on Machine Learning, ICML 2013, pp. 396–404 (2013)
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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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