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Perfect clustering for stochastic blockmodel graphs via adjacency spectral embedding. (English) Zbl 1308.62131
Summary: Vertex clustering in a stochastic blockmodel graph has wide applicability and has been the subject of extensive research. In this paper, we provide a short proof that the adjacency spectral embedding can be used to obtain perfect clustering for the stochastic blockmodel and the degree-corrected stochastic blockmodel. We also show an analogous result for the more general random dot product graph model.

MSC:
62H30 Classification and discrimination; cluster analysis (statistical aspects)
05C80 Random graphs (graph-theoretic aspects)
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[1] Athreya, A., Lyzinski, V., Marchette, D. J., Priebe, C. E., Sussman, D. L., and Tang, M., A limit theorem for scaled eigenvectors of random dot product graphs., Sankhya A , accepted for publication. · Zbl 1338.62061
[2] Bickel, P. J. and Chen, A., A nonparametric view of network models and Newman-Girvan and other modularities., Proc. Natl. Acad. Sci. USA , 106 :21068-21073, 2009. · Zbl 1359.62411
[3] Bickel, P. J., Chen, A., and Levina, E., The method of moments and degree distributions for network models., Ann. Statist. , 39:38-59, 2011. · Zbl 1232.91577
[4] Chaudhuri, K., Chung, F., and Tsiatas, A., Spectral partitioning of graphs with general degrees and the extended planted partition model. In, Proceedings of the 25th Conference on Learning Theory , 2012.
[5] Choi, D. S., Wolfe, P. J., and Airoldi, E. M., Stochastic blockmodels with a growing number of classes., Biometrika , 99:273-284, 2012. · Zbl 1318.62207
[6] Choi, D. and Wolfe, P. J., Co-clustering separately exchangeable network data., Ann. Statist. , 42(1):29-63, 2014. · Zbl 1294.62059
[7] Hoff, P. D., Raftery, A. E., and Handcock, M. S., Latent space approaches to social network analysis., J. Amer. Statist. Assoc. , 97(460) :1090-1098, 2002. · Zbl 1041.62098
[8] Holland, P. W., Laskey, K., and Leinhardt, S., Stochastic blockmodels: First steps., Social Networks , 5:109-137, 1983.
[9] Karrer, B. and Newman, M. E. J., Stochastic blockmodels and community structure in networks., Phys. Rev. E , 83, 2011.
[10] Lei, J. and Rinaldo, A., Consistency of spectral clustering in stochastic blockmodels., Ann. Statist. , 43:215-237, 2015. · Zbl 1308.62041
[11] Lyzinski, V., Sussman, D. L., Fishkind, D. E., Pao, H., Chen, L., Vogelstein, J. T., and Priebe, C. E., Spectral clustering for divide-and-conquer graph matching. Arxiv preprint at http://arxiv.org/abs/ 1310.1297, 2014.
[12] Mossel, E., Neeman, J., and Sly, A., Consistency thresholds for binary symmetric block models. ArXiv preprint at http://arxiv.org/abs/ 1407.1591, 2014. · Zbl 1336.05117
[13] Mossel, E., Neeman, J., and Sly, A., Stochastic block models and reconstruction., Probab. Theory Related Fields , · Zbl 1350.05154
[14] Newman, M. E. J., Modularity and community structure in networks., Proc. Natl. Acad. Sci. USA , 103(23) :8577-8582, 2006.
[15] Oliveira, R. I., Concentration of the adjacency matrix and of the laplacian in random graphs with independent edges. Arxiv preprint at http://arxiv.org/abs /0911.0600, 2010.
[16] Pollard, D., Strong consistency of \(k\)-means clustering., Ann. Statist. , 9:135-140, 1981. · Zbl 0451.62048
[17] Qin, T. and Rohe, K., Regularized spectral clustering under the degree-corrected stochastic blockmodel., NIPS , 2013.
[18] Rohe, K., Chatterjee, S., and Yu, B., Spectral clustering and the high-dimensional stochastic blockmodel., Ann. Statist. , 39 :1878-1915, 2011. · Zbl 1227.62042
[19] Snijders, T. A. B. and Nowicki, K., Estimation and Prediction for Stochastic Blockmodels for Graphs with Latent Block Structure., J. Classification , 14:75-100, 1997. · Zbl 0896.62063
[20] Sussman, D. L., Tang, M., Fishkind, D. E., and Priebe, C. E., A consistent adjacency spectral embedding for stochastic blockmodel graphs., J. Amer. Statist. Assoc. , 107(499) :1119-1128, 2012. · Zbl 1443.62188
[21] Sussman, D. L., Tang, M., and Priebe, C. E., Consistent latent position estimation and vertex classification for random dot product graphs., IEEE Trans. Pattern. Anal. , 36:48-57, 2014.
[22] Suwan, S., Lee, D. S., Tang, R., Sussman, D. L., Tang, M., and Priebe, C. E., Empirical bayes estimation for the stochastic blockmodel. Arxiv preprint at arxiv.org/abs /1405.6070, 2014. · Zbl 1333.62088
[23] Tang, M., Sussman, D. L., and Priebe, C. E., Universally consistent vertex classification for latent position graphs., Ann. Statist. , 41 :1406-1430, 2013. · Zbl 1273.62147
[24] Tang, M., Athreya, A., Sussman, D. L., Lyzinski, V., and Priebe, C. E., A semiparametric two-sample hypothesis testing problem for random dot product graphs. ArXiv preprint at http://arxiv.org/abs/ 1403.7249, 2014. · Zbl 1308.62131
[25] Tropp, J. A., Freedman’s inequality for matrix martingales., Electron. Commun. Probab , 16:262-270, 2011. · Zbl 1225.60017
[26] Wolfe, P. J. and Olhede, S. C., Nonparametric graphon estimation. ArXiv preprint at http://arxiv.org/abs /1309/5936, 2013.
[27] Young, S. and Scheinerman, E., Random dot product graph models for social networks. In, Proceedings of the 5th International Conference on Algorithms and Models for the Web-Graph , pages 138-149, 2007. · Zbl 1136.05322
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