Abstract
We derive the limiting distribution for the outlier eigenvalues of the adjacency matrix for random graphs with independent edges whose edge probability matrices have low-rank structure. We show that when the number of vertices tends to infinity, the leading eigenvalues in magnitude are jointly multivariate Gaussian with bounded covariances. As a special case, this implies a limiting normal distribution for the outlier eigenvalues of stochastic blockmodel graphs and their degree-corrected or mixed-membership variants. Our result extends the classical result of Füredi and Komlós on the fluctuation of the largest eigenvalue for Erdős–Rényi graphs.
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References
Airoldi, E. M., Blei, D. M., Fienberg, S. E. and Xing, E. P. (2008). Mixed membership stochastic blockmodels. Journal of Machine Learning Research9, 1981–2014.
Arnold, L. (1967). On the asymptotic distribution of the eigenvalues of random matrices. Journal of Mathematical Analysis and Applications, 20, 262–268.
Avrachenkov, K., Cottatellucci, L. and Kadavankandy, A. (2015). Spectral properties of random matrices for stochastic block model. In Proceedings of the 4th, International Workshop on Physics-Inspired Paradigms in Wireless Communications and Networks (pp. 537–544).
Benaych-Georges, F. and Nadakuditi, R. R. (2011). The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. Advances in Mathematics 227, 494–521.
Bordenave, C. and Capitaine, M. (2016). Outlier eigenvalues for deformed i.i.d. random matrices. Communications on Pure and Applied Mathematics 69, 2131–2194.
Boucheron, S., Lugosi, G. and Massart, P. (2003). Concentration inequalities using the entropy method. Annals of Probability 31, 1583–1614.
Cape, J., Tang, M. and Priebe, C. E. (2017). The Kato-Temple inequality and eigenvalue concentration. Electronic Journal of Statistics 11, 3954–3978.
Capitaine, M., Donati-Martin, C. and Féral, D. (2009). The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations. Ann. Probab. 37, 1–47.
Capitaine, M., Donati-Martin, C. and Féral, D. (2012). Central limit theorems for eigenvalues of deformations of Wigner matrices. Annales de l’Institut Henri Poincaré - Probabilités et Statistiques 48, 107–133.
Chakrabarty, A., Chakraborty, S. and Hazra, R. S. (2020). Eigenvalues outside the bulk of inhomogeneous Erdos-Renyi random graphs. Journal of Statistical Physics 181, 1746–1780.
Davis, C. and Kahan, W. (1970). The rotation of eigenvectors by a pertubation. III. SIAM Journal on Numerical Analysis, 7, 1–46.
Ding, X. and Jiang, T. (2010). Spectral distributions of adjacency and Laplacian matrices of random graphs. Annals of Applied Probability 20, 2086–2117.
Donoho, D., Gavish, M. and Johnstone, I. (2018). Optimal shrinkage of eigenvalues in the spiked covariance model. Annals of Statistics 46, 1742–1778.
Erdös, L., Péché, S., Ramirez, J. A., Schlein, B. and Yau, H.-T. (2010). Bulk universality for W,igner matrices. Communications on Pure and Applied Mathematics 63, 895–925.
Erdös, L., Schlein, B. and Yau, H. -T. (2009). Local semicircle law and complete delocalization for Wigner random matrices. Communications in Mathematical Physics 287, 641–655.
Füredi, Z. and Komlós, J. (1981). The eigenvalues of random symmetric matrices. Combinatorica 1, 233–241.
Ghoshdastidar, D., Gutzeit, M., Carpentier, A. and von Luxburg, U. (2017). Two-sample tests for large random graphs using network statistics. In Proceedings of Machine Learning Research (vol. 65, pp. 1–24).
Holland, P. W., Laskey, K. B. and Leinhardt, S. (1983). Stochastic blockmodels: first steps. Social Networks 5, 109–137.
Karrer, B. and Newman, M. E. J. (2011). Stochastic blockmodels and community structure in networks. Physical Review E 83, 016107.
Knowles, A. and Yin, J. (2013). Eigenvector distribution of Wigner matrices. Probability Theory and Related Fields 155, 543–582.
Knowles, A. and Yin, J. (2014). The outliers of a deformed Wigner matrix. Annals of Probability 42, 1980–2031.
Koltchinskii, V. and Giné, E. (2000). Random matrix approximation of spectra of integral operators. Bernoulli 6, 113–167.
Lei, J. (2016). A goodness-of-fit test for stochastic block models. Annals of Statistics 44, 401–424.
Noroozi, M., Rimal, R. and Pensky, M. (2019). Estimation and clustering in popularity adjusted stochastic blockmodel. arXiv:http://arxiv.org/abs/1902.00431
O’Rourke, S. and Renfrew, D. (2014). Low rank perturbation of large elliptic random matrices. Electronic Journal of Probability 19, 1–65.
O’Rourke, S. and Vu, V. (2018). K. Wang. Random perturbation of low rank matrices Improving classical bounds. Linear Algebra and its Applications540, 26–59.
Péché, S. (2006). The largest eigenvalue of small-rank perturbations of Hermitean random matrices. Probability Theory and Related Fields 134, 127–173.
Pizzo, A., Renfrew, D. and Soshnikov, A. (2013). On finite rank deformation of Wigner matrices. Annales de l’Institut Henri Poincaré Probabilités et Statistiques 49, 64–94.
Rubin-Delanchy, P., Cape, J., Priebe, C. E. and Tang, M. (2017). A statistical interpretation of spectral embedding The random dot product graph. arXiv:http://arxiv.org/abs/1709.05506
Sengupta, S. and Chen, Y. (2018). A blockmodel for node popularity in networks with community structure. Journal of the Royal Statistical Society Series B, 365–386.
Soshnikov, A. (1999). Universality at the edge of the spectrum in Wigner random matrices. Communications in Mathematical Physics 207, 697–733.
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. Journal of Computational and Graphical Statistics 26, 344–354.
Tao, T. and Vu, V. (2010). Random matrices: universality of local eigenvalue statistics up to the edge. Communications in Mathematical Physics 298, 549–572.
Tao, T. and Vu, V. (2012). Random matrices: universal properties of eigenvectors. Random Matrices: Theory and Applications, 1.
Tokuda, T. (2018). Statistical test for detecting community structure in real-valued edge-weighted graphs. PLoS ONE 13, e0194079.
Wigner, E. P. (1955). Characteristic vectors of bordered matrices with infinite dimensions. Annals of Mathematics 62, 548–564.
Young, S. and Scheinerman, E. (2007). Random dot product graph models for social networks. In Proceedings of the 5th international conference on algorithms and models for the web-graph (pp. 138–149).
Yu, Y., Wang, T. and Samworth, R. J. (2015). A useful variant of the Davis-Kahan theorem for statisticians. Biometrika 102, 315–323.
Zhang, X., Nadakuditi, R. R. and Newman, M. E. (2014). Spectra of random graphs with community structure and arbitrary degrees. Physical Review E, 89.
Funding
The research leading to these results received funding from the Defense Advanced Research Projects Agency under Grant Agreement No. FA8750-12-2-0303.
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Athreya, A., Cape, J. & Tang, M. Eigenvalues of Stochastic Blockmodel Graphs and Random Graphs with Low-Rank Edge Probability Matrices. Sankhya A 84, 36–63 (2022). https://doi.org/10.1007/s13171-021-00268-x
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DOI: https://doi.org/10.1007/s13171-021-00268-x