Annals of Statistics

Limit theorems for eigenvectors of the normalized Laplacian for random graphs

Minh Tang and Carey E. Priebe

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We prove a central limit theorem for the components of the eigenvectors corresponding to the $d$ largest eigenvalues of the normalized Laplacian matrix of a finite dimensional random dot product graph. As a corollary, we show that for stochastic blockmodel graphs, the rows of the spectral embedding of the normalized Laplacian converge to multivariate normals and, furthermore, the mean and the covariance matrix of each row are functions of the associated vertex’s block membership. Together with prior results for the eigenvectors of the adjacency matrix, we then compare, via the Chernoff information between multivariate normal distributions, how the choice of embedding method impacts subsequent inference. We demonstrate that neither embedding method dominates with respect to the inference task of recovering the latent block assignments.

Article information

Ann. Statist., Volume 46, Number 5 (2018), 2360-2415.

Received: August 2016
Revised: June 2017
First available in Project Euclid: 17 August 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation
Secondary: 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20] 62B10: Information-theoretic topics [See also 94A17]

Spectral clustering random dot product graph stochastic blockmodels convergence of eigenvectors Chernoff information


Tang, Minh; Priebe, Carey E. Limit theorems for eigenvectors of the normalized Laplacian for random graphs. Ann. Statist. 46 (2018), no. 5, 2360--2415. doi:10.1214/17-AOS1623.

Export citation


  • [1] Ali, S. M. and Silvey, S. D. (1966). A general class of coefficients of divergence of one distribution from another. J. Roy. Statist. Soc. Ser. B 28 131–142.
  • [2] Amini, A., Chen, A., Bickel, P. and Levina, E. (2013). Pseudo-likelihood methods for community detection in large sparse networks. Ann. Statist. 41 2097–2122.
  • [3] Asta, D. and Shalizi, C. (2014). Geometric network comparison. arXiv preprint at arXiv:1411.1350.
  • [4] Athreya, A., Lyzinski, V., Marchette, D. J., Priebe, C. E., Sussman, D. L. and Tang, M. (2016). A limit theorem for scaled eigenvectors of random dot product graphs. Sankhya A 78 1–18.
  • [5] Belkin, M. and Niyogi, P. (2003). Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput. 15 1373–1396.
  • [6] 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.
  • [7] Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31 3–122.
  • [8] Boucheron, S., Lugosi, G. and Massart, P. (2003). Concentration inequalities using the entropy method. Ann. Probab. 31 1583–1614.
  • [9] Chaudhuri, K., Chung, F. and Tsiatas, A. (2012). Spectral partitioning of graphs with general degrees and the extended planted partition model. In Proceedings of the 25th Conference on Learning Theory.
  • [10] Chernoff, H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Stat. 23 493–507.
  • [11] Chernoff, H. (1956). Large sample theory: Parametric case. Ann. Math. Stat. 27 1–22.
  • [12] Choi, D. S., Wolfe, P. J. and Airoldi, E. M. (2012). Stochastic blockmodels with a growing number of classes. Biometrika 99 273–284.
  • [13] Chung, F. R. K. (1997). Spectral Graph Theory 92. Amer. Math. Soc.
  • [14] Coifman, R. and Lafon, S. (2006). Diffusion maps. Appl. Comput. Harmon. Anal. 21 5–30.
  • [15] Csizár, I. (1967). Information-type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hungar. 2 229–318.
  • [16] Diaconis, P. and Janson, S. (2008). Graph limits and exchangeable random graphs. Rend. Mat. Appl. (7) 28 33–61.
  • [17] Fraley, C. and Raftery, A. E. (1999). MCLUST: Software for model-based cluster analysis. J. Classification 16 297–306.
  • [18] 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.
  • [19] Holland, P. W., Laskey, K. B. and Leinhardt, S. (1983). Stochastic blockmodels: First steps. Soc. Netw. 5 109–137.
  • [20] Joseph, A. and Yu, B. (2016). Impact of regularization on spectral clustering. Ann. Statist. 44 1765–1791.
  • [21] Karrer, B. and Newman, M. E. J. (2011). Stochastic blockmodels and community structure in networks. Phys. Rev. E (3) 83 016107.
  • [22] Leang, C. C. and Johnson, D. H. (1997). On the asymptotics of M-hypothesis Bayesian detection. IEEE Trans. Inform. Theory 43 280–282.
  • [23] Lei, J. and Rinaldo, A. (2015). Consistency of spectral clustering in stochastic block models. Ann. Statist. 43 215–237.
  • [24] Liese, F. and Vajda, I. (2006). On divergences and informations in statistics and information theory. IEEE Trans. Inform. Theory 52 4394–4412.
  • [25] Lu, L. and Peng, X. (2013). Spectra of edge-independent random graphs. Electron. J. Combin. 20.
  • [26] Luxburg, U. V. (2007). A tutorial on spectral clustering. Stat. Comput. 17 395–416.
  • [27] Lyzinski, V., Sussman, D. L., Tang, M., Athreya, A. and Priebe, C. E. (2014). Perfect clustering for stochastic blockmodel graphs via adjacency spectral embedding. Electron. J. Stat. 8 2905–2922.
  • [28] McSherry, F. (2001). Spectral partitioning of random graphs. In Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science 529–537.
  • [29] Merris, R. (1994). Laplacian matrices of graphs: A survey. Linear Algebra Appl. 197 143–176.
  • [30] Mossel, E., Neeman, J. and Sly, A. (2015). Reconstruction and estimation in the planted partition model. Probab. Theory Related Fields 162 431–461.
  • [31] Nickel, C. L. M. (2006). Random dot product graphs: A model for social networks. Ph.D. thesis, Johns Hopkins Univ., Baltimore, MD.
  • [32] Oliveira, R. I. (2009). Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges. arXiv preprint at arXiv:0911.0600.
  • [33] Qin, T. and Rohe, K. (2013). Regularized spectral clustering under the degree-corrected stochastic blockmodel. In NIPS.
  • [34] Rohe, K., Chatterjee, S. and Yu, B. (2011). Spectral clustering and the high-dimensional stochastic blockmodel. Ann. Statist. 39 1878–1915.
  • [35] Sarkar, P. and Bickel, P. J. (2015). Role of normalization for spectral clustering in stochastic blockmodels. Ann. Statist. 43 962–990.
  • [36] Shi, J. and Malik, J. (2000). Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22 888–905.
  • [37] Snijders, T. A. B. and Nowicki, K. (1997). Estimation and prediction for stochastic blockmodels for graphs with latent block structure. J. Classification 14 75–100.
  • [38] Stewart, G. W. and Sun, J. (1990). Matrix Pertubation Theory. Academic Press, New York.
  • [39] Sussman, D. L. (2014). Foundations of adjacency spectral embedding. PhD Thesis, Johns Hopkins Univ., Baltimore, MD.
  • [40] Sussman, D. L., Tang, M., Fishkind, D. E. and Priebe, C. E. (2012). A consistent adjacency spectral embedding for stochastic blockmodel graphs. J. Amer. Statist. Assoc. 107 1119–1128.
  • [41] Tang, M., Athreya, A., Sussman, D. L., Lyzinski, V., Park, Y. and Priebe, C. E. (2017). A semiparametric two-sample hypothesis testing problem for random dot product graphs. J. Comput. Graph. Statist. 26 344–354.
  • [42] Tang, M., Athreya, A., Sussman, D. L., Lyzinski, V. and Priebe, C. E. (2017). A nonparametric two-sample hypothesis testing problem for random graphs. Bernoulli 23 1599–1630.
  • [43] von Luxburg, U., Belkin, M. and Bousquet, O. (2008). Consistency of spectral clustering. Ann. Statist. 36 555–586.
  • [44] Wolfe, P. J. and Olhede, S. C. (2013). Nonparametric graphon estimation. arXiv preprint at arXiv:1309.5936.
  • [45] Yang, J. J., Han, Q. and Airoldi, E. M. (2014). Nonparametric estimation and testing of exchangeable graph models. In Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics 1060–1067.