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Spectral Clustering and Block Models: A Review and a New Algorithm

  • Sharmodeep BhattacharyyaEmail author
  • Peter J. Bickel
Conference paper
Part of the Abel Symposia book series (ABEL, volume 11)

Abstract

We focus on spectral clustering of unlabeled graphs and review some results on clustering methods which achieve weak or strong consistent identification in data generated by such models. We also present a new algorithm which appears to perform optimally both theoretically using asymptotic theory.

References

  1. 1.
    Abbe, E., Bandeira, A.S., Hall, G.: Exact recovery in the stochastic block model (2014). arXiv preprint arXiv:1405.3267Google Scholar
  2. 2.
    Amini, A.A., Levina, E.: On semidefinite relaxations for the block model (2014). arXiv preprint arXiv:1406.5647Google Scholar
  3. 3.
    Amini, A.A., Chen, A., Bickel, P.J., Levina, E.: Pseudo-likelihood methods for community detection in large sparse networks. Ann. Stat. 41(4), 2097–2122 (2013). doi: 10.1214/13-AOS1138. http://dx.doi.org/10.1214/13-AOS1138
  4. 4.
    Bhamidi, S., Van der Hofstad, R., Hooghiemstra, G.: First passage percolation on the Erds-Renyi random graph. Comb. Probab. Comput. 20(5), 683–707 (2011)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bhattacharyya, S., Bickel, P.J.: Community detection in networks using graph distance (2014). arXiv preprint arXiv:1401.3915Google Scholar
  6. 6.
    Bickel, P.J., Chen, A.: A nonparametric view of network models and Newman–Girvan and other modularities. Proc. Natl. Acad. Sci. 106(50), 21068–21073 (2009)CrossRefGoogle Scholar
  7. 7.
    Bickel, P., Choi, D., Chang, X., Zhang, H.: Asymptotic normality of maximum likelihood and its variational approximation for stochastic blockmodels. Ann. Stat. 41(4), 1922–1943 (2013). doi: 10.1214/13-AOS1124. http://dx.doi.org/10.1214/13-AOS1124
  8. 8.
    Bollobás, B., Janson, S., Riordan, O.: The phase transition in inhomogeneous random graphs. Random Struct. Algorithm. 31(1), 3–122 (2007)CrossRefzbMATHGoogle Scholar
  9. 9.
    Bordenave, C., Lelarge, M., Massoulié, L.: Non-backtracking spectrum of random graphs: community detection and non-regular Ramanujan graphs (2015). arXiv preprint arXiv:1501.06087Google Scholar
  10. 10.
    Celisse, A., Daudin, J.J., Pierre, L.: Consistency of maximum-likelihood and variational estimators in the stochastic block model. Electron. J. Stat. 6, 1847–1899 (2012). doi: 10.1214/12-EJS729. http://dx.doi.org/10.1214/12-EJS729
  11. 11.
    Chatelin, F.: Spectral Approximation of Linear Operators. SIAM, Philadelphia (1983)zbMATHGoogle Scholar
  12. 12.
    Davis, C., Kahan, W.M.: The rotation of eigenvectors by a perturbation. III. SIAM J. Numer. Anal. 7(1), 1–46 (1970)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Decelle, A., Krzakala, F., Moore, C., Zdeborová, L.: Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications. Phys. Rev. E 84(6), 066106 (2011)CrossRefGoogle Scholar
  14. 14.
    Fiedler, M.: Algebraic connectivity of graphs. Czechoslov. Math. J. 23(98), 298–305 (1973)MathSciNetGoogle Scholar
  15. 15.
    Floyd, R.W.: Algorithm 97: shortest path. Commun. ACM 5(6), 345 (1962)CrossRefGoogle Scholar
  16. 16.
    Gao, C., Ma, Z., Zhang, A.Y., Zhou, H.H.: Achieving optimal misclassification proportion in stochastic block model (2015). arXiv preprint arXiv:1505.03772Google Scholar
  17. 17.
    Girvan, M., Newman, M.E.: Community structure in social and biological networks. Proc. Natl. Acad. Sci. 99(12), 7821–7826 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Hartigan, J.A.: Clustering Algorithms. Wiley Series in Probability and Mathematical Statistics. Wiley, New York/London/Sydney (1975)zbMATHGoogle Scholar
  19. 19.
    Holland, P.W., Laskey, K.B., Leinhardt, S.: Stochastic blockmodels: first steps. Soc. Netw. 5(2), 109–137 (1983)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Johnson, D.B.: Efficient algorithms for shortest paths in sparse networks. J. ACM 24(1), 1–13 (1977)CrossRefzbMATHGoogle Scholar
  21. 21.
    Katō, T.: Perturbation Theory for Linear Operators, vol. 132. Springer, Berlin (1995)zbMATHGoogle Scholar
  22. 22.
    Massoulié, L.: Community detection thresholds and the weak Ramanujan property. In: Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pp. 694–703. ACM, New York (2014)Google Scholar
  23. 23.
    Mode, C.J.: Multitype Branching Processes: Theory and Applications, vol. 34. American Elsevier Pub. Co., New York (1971)zbMATHGoogle Scholar
  24. 24.
    Mossel, E., Neeman, J., Sly, A.: Stochastic block models and reconstruction (2012). arXiv preprint arXiv:1202.1499Google Scholar
  25. 25.
    Mossel, E., Neeman, J., Sly, A.: A proof of the block model threshold conjecture (2013). arXiv preprint arXiv:1311.4115Google Scholar
  26. 26.
    Ng, A.Y., Jordan, M.I., Weiss, Y., et al.: On spectral clustering: analysis and an algorithm. Adv. Neural Inf. Process. Syst. 2, 849–856 (2002)Google Scholar
  27. 27.
    Rohe, K., Chatterjee, S., Yu, B.: Spectral clustering and the high-dimensional stochastic blockmodel. Ann. Stat. 39(4), 1878–1915 (2011). doi: 10.1214/11-AOS887. http://dx.doi.org/10.1214/11-AOS887
  28. 28.
    Rousseeuw, P.J., Leroy, A.M.: Robust Regression and Outlier Detection. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Wiley, New York (1987). doi: 10.1002/0471725382. http://dx.doi.org/10.1002/0471725382
  29. 29.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22(8), 888–905 (2000)CrossRefGoogle Scholar
  30. 30.
    Sussman, D.L., Tang, M., Fishkind, D.E., Priebe, C.E.: A consistent adjacency spectral embedding for stochastic blockmodel graphs. J. Am. Stat. Assoc. 107(499), 1119–1128 (2012). doi: 10.1080/01621459.2012.699795. http://dx.doi.org/10.1080/01621459.2012.699795
  31. 31.
    von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007)CrossRefMathSciNetGoogle Scholar
  32. 32.
    von Luxburg, U., Belkin, M., Bousquet, O.: Consistency of spectral clustering. Ann. Stat. 36(2), 555–586 (2008). doi: 10.1214/009053607000000640. http://dx.doi.org/10.1214/009053607000000640
  33. 33.
    Warshall, S.: A theorem on boolean matrices. J. ACM 9(1), 11–12 (1962)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of StatisticsOregon State UniversityCorvallisUSA
  2. 2.Department of StatisticsUniversity of California at BerkeleyBerkeleyUSA

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