The Annals of Statistics

Co-clustering separately exchangeable network data

David Choi and Patrick J. Wolfe

Full-text: Open access

Abstract

This article establishes the performance of stochastic blockmodels in addressing the co-clustering problem of partitioning a binary array into subsets, assuming only that the data are generated by a nonparametric process satisfying the condition of separate exchangeability. We provide oracle inequalities with rate of convergence $\mathcal{O}_{P}(n^{-1/4})$ corresponding to profile likelihood maximization and mean-square error minimization, and show that the blockmodel can be interpreted in this setting as an optimal piecewise-constant approximation to the generative nonparametric model. We also show for large sample sizes that the detection of co-clusters in such data indicates with high probability the existence of co-clusters of equal size and asymptotically equivalent connectivity in the underlying generative process.

Article information

Source
Ann. Statist., Volume 42, Number 1 (2014), 29-63.

Dates
First available in Project Euclid: 15 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.aos/1389795744

Digital Object Identifier
doi:10.1214/13-AOS1173

Mathematical Reviews number (MathSciNet)
MR3161460

Zentralblatt MATH identifier
1294.62059

Subjects
Primary: 62G05: Estimation
Secondary: 05C80: Random graphs [See also 60B20] 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Keywords
Bipartite graph network clustering oracle inequality profile likelihood statistical network analysis stochastic blockmodel and co-blockmodel

Citation

Choi, David; Wolfe, Patrick J. Co-clustering separately exchangeable network data. Ann. Statist. 42 (2014), no. 1, 29--63. doi:10.1214/13-AOS1173. https://projecteuclid.org/euclid.aos/1389795744


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References

  • Airoldi, E. M., Blei, D. M., Fienberg, S. E. and Xing, E. P. (2008). Mixed membership stochastic blockmodels. J. Mach. Learn. Res. 9 1981–2014.
  • Alon, N., Fernandez de la Vega, W., Kannan, R. and Karpinski, M. (2003). Random sampling and approximation of MAX-CSPs. J. Comput. System Sci. 67 212–243.
  • Bickel, P. J. and Chen, A. (2009). A nonparametric view of network models and Newman–Girvan and other modularities. Proc. Natl. Acad. Sci. USA 106 21068–21073.
  • Bickel, P. J., Chen, A. and Levina, E. (2011). The method of moments and degree distributions for network models. Ann. Statist. 39 2280–2301.
  • Borgs, C., Chayes, J., Lovász, L., Sós, V. T., Szegedy, B. and Vesztergombi, K. (2006). Graph limits and parameter testing. In STOC’06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing 261–270. ACM, New York.
  • Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2008). Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing. Adv. Math. 219 1801–1851.
  • Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2012). Convergent sequences of dense graphs. II. Multiway cuts and statistical physics. Ann. of Math. (2) 176 151–219.
  • Bousquet, O., Boucheron, S. and Lugosi, G. (2004). Introduction to statistical learning theory. In Advanced Lectures on Machine Learning (O. Bousquet, U. von Luxburg and G. Rätsch, eds.) 169–207. Springer, Berlin.
  • Chatterjee, S. (2012). Matrix estimation by universal singular value thresholding. Preprint. Available at arXiv:1212.1247.
  • Choi, D. S., Wolfe, P. J. and Airoldi, E. M. (2012). Stochastic blockmodels with a growing number of classes. Biometrika 99 273–284.
  • Clémençon, S., Lugosi, G. and Vayatis, N. (2008). Ranking and empirical minimization of $U$-statistics. Ann. Statist. 36 844–874.
  • Diaconis, P. and Janson, S. (2008). Graph limits and exchangeable random graphs. Rend. Mat. Appl. (7) 28 33–61.
  • Fienberg, S. E. (2012). A brief history of statistical models for network analysis and open challenges. J. Comput. Graph. Statist. 21 825–839.
  • Fishkind, D. E., Sussman, D. L., Tang, M., Vogelstein, J. T. and Priebe, C. E. (2013). Consistent adjacency-spectral partitioning for the stochastic block model when the model parameters are unknown. SIAM J. Matrix Anal. Appl. 34 23–39.
  • Flynn, C. J. and Perry, P. O. (2012). Consistent biclustering. Preprint. Available at arXiv:1206.6927.
  • Fortunato, S. and Barthélemy, M. (2007). Resolution limit in community detection. Proc. Natl. Acad. Sci. USA 104 36–41.
  • Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.
  • Hoff, P. D. (2009). Multiplicative latent factor models for description and prediction of social networks. Computat. Math. Org. Theory 15 261–272.
  • Hoff, P. D., Raftery, A. E. and Handcock, M. S. (2002). Latent space approaches to social network analysis. J. Amer. Statist. Assoc. 97 1090–1098.
  • Kim, M. and Leskovec, J. (2012). Multiplicative attribute graph model of real-world networks. Internet Math. 8 113–160.
  • Miller, K. T., Griffiths, T. L. and Jordan, M. I. (2009). Nonparametric latent feature models for link prediction. In Advances in Neural Information Processing Systems 22 (Y. Bengio, D. Schuurmans, J. Lafferty, C. K. I. Williams and A. Culotta, eds.) 1276–1284. MIT Press, Cambridge, MA.
  • Newman, M. E. J. (2006). Modularity and community structure in networks. Proc. Natl. Acad. Sci. USA 103 8577–8582.
  • Rohe, K., Chatterjee, S. and Yu, B. (2011). Spectral clustering and the high-dimensional stochastic blockmodel. Ann. Statist. 39 1878–1915.
  • Rohe, K. and Yu, B. (2012). Co-clustering for directed graphs: The stochastic co-blockmodel and a spectral algorithm. Preprint. Available at arXiv:1204.2296.
  • Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge.
  • White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica 50 1–25.
  • Zhao, Y., Levina, E. and Zhu, J. (2011). Community extraction for social networks. Proc. Natl. Acad. Sci. USA 108 7321–7326.
  • Zhao, Y., Levina, E. and Zhu, J. (2012). Consistency of community detection in networks under degree-corrected stochastic block models. Ann. Statist. 40 2266–2292.