A limit theorem for scaled eigenvectors of random dot product graphs

Sankhya: The Indian Journal of Statistics

Volumn 78A, Issue , 2016, Pages 1-18

  Athreya, A ^a   Priebe, C E ^a   Tang, M ^a   Lyzinski, V ^a   Marchette, D J ^b   Sussman, D L ^c  

^a Johns Hopkins University   (United States)

^b NAVAL SURFACE WARFARE CENTER   (United States)

^c HARVARD UNIVERSITY   (United States)

Author keywords

Central limit theorem; Model based clustering; Random dot product graph

Indexed keywords

EID: 84973351119     PISSN: 09727671     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (31)

1. ALDOUS, D.J. (1981). Representations for partially exchangeable arrays of random variables. J. Multivariate. Anal. 11, 581-598.
2. 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, 21, 068-73.
3. BICKEL, P.J., CHEN, A. and LEVINA, E. (2011). The method of moments and degree distributions for network models. Ann. Statist. 39, 38-59.
4. CHOI, D.S., WOLFE, P.J. and AIROLDI, E.M. (2012). Stochastic blockmodels with a growing number of classes. Biometrika 99, 273-284.
5. CHUNG, F.R.K. (1997). Spectral Graph Theory. American Mathematical Society.
6. DIACONIS, P. and JANSON, S. (2008). Graph limits and exchangeable random graphs. Rend. Mat. Appl. 28, 33-61.
7. FIEDLER, M. (1973). Algebraic connectivity of graphs. Czechoslovak Math. J 23, 298-305.
8. 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.
9. FORTUNATO, S. (2010). Community detection in graphs. Phys. Rep. 486, 75-174.
10. FRALEY, C. and RAFTERY, A.E. (1999). MCLUST: Software for model-based cluster analysis. J. Classif. 16, 297-306.
11. FüREDI, Z. and KOMLóS, J. (1981). The eigenvalues of random symmetric matrices. Combinatorica 1, 233-241.
12. GOLDENBERG, A., ZHENG, A.X., FIENBERG, S.E. and AIROLDI, E.M. (2010). A survey of statistical network models. Foundations and Trends® in Machine Learning 2, 129-233.
13. HOFF, P.D., RAFTERY, A.E. and HANDCOCK, M.S. (2002). Latent space approaches to social network analysis. J. Am. Statist. Assoc. 97, 1090-1098.
14. HOOVER, D.N. (1979). Relations on probability spaces and arrays of random variables. Tech. rep. Institute for Advanced Study.
15. JANSON, S. (2005). The first eigenvalue of random graphs. Comb. Probab. Comput. 14, 815-828.
16. KNOWLES, A. and YIN, J. (2011). Eigenvector distribution of Wigner matrices. Probab. Theory Related Fields, 1-40.
17. KRIVELEVICH, M. and SUDAKOV, B. (2003). The largest eigenvalue of sparse random graphs. Comb. Probab. Comput. 12, 61-72.
18. LUXBURG, U.V. (2007). A tutorial on spectral clustering. Stat. Comput. 17, 395-416.
19. MARCHETTE, D., PRIEBE, C.E. and COPPERSMITH, G. (2011). Vertex nomination via attributed random dot product graphs. In Proceedings of the 57th ISI World Statistic Congress.
20. OLIVEIRA, R.I. (2010). Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges. arXiv:preprint.
21. ROHE, K., CHATTERJEE, S. and YU, B. (2011). Spectral clustering and the high-dimensional stochastic blockmodel. Ann. Statist. 39, 1878-1915.
22. SARKAR, P. and BICKEL, P.J. (2015). Role of normalization in spectral clustering for stochastic blockmodels. Ann. Statist. 43, 962-990.
23. SILVERSTEIN, J.W. (1994). The spectral radii and norms of large-dimensional non-central random matrices. Comm. Statist. Stochastic Models 10, 3, 525-532.
24. SUSSMAN, D.L. (2014). Foundations of Adjacency Spectral Embedding. PhD Thesis, Johns Hopkins University.
25. SUSSMAN, D.L., TANG, M., FISHKIND, D.E. and PRIEBE, C.E. (2012). A consistent adjacency spectral embedding for stochastic blockmodel graphs. J. Am. Statist. Assoc 107, 1119-1128.
26. SUSSMAN, D.L., TANG, M. and PRIEBE, C.E. (2014). Consistent latent position estimation and vertex classification for random dot product graphs. IEEE T. Pattern. Anal. 36, 48-57.
27. TANG, M., SUSSMAN, D.L. and PRIEBE, C.E. (2013). Universally consistent vertex classification for latent position graphs. Ann. Statist. 41, 1406-1430.
28. TAO, T. and VU, V. (2012). Random matrices: Universal properties of eigenvectors. Random Matrices: Theory and Applications 1.
29. TROPP, J.A. (2011). Freedmans inequality for matrix martingales. Electron. Commun. Probab. 16, 262-270.
30. YAN, T. and XU, J. (2013). A central limit thereom in the β-model for undirected random graphs with a diverging number of vertices. Biometrika 100, 519-524.
31. YOUNG, S. and SCHEINERMAN, E. (2007). Random dot product graph models for social networks. In Algorithms and models for the web-graph. Springer, p. 138-149.