Spectral clustering and the high-dimensional stochastic blockmodel

Annals of Statistics

Volumn 39, Issue 4, 2011, Pages 1878-1915

  Rohe, Karl ^a   Chatterjee, Sourav ^a   Yu, Bin ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Clustering; Convergence of eigenvectors; Latent space model; Principal components analysis.; Spectral clustering; Stochastic Blockmodel

Indexed keywords

EID: 80052860256     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/11-AOS887     Document Type: Review

References (57)

1. 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.
2. ALBERT, R. and BARABÁSI, A. L. (2002). Statistical mechanics of complex networks. Rev. Modern Phys. 74 47-97. MR1895096
3. BARABÁSI, A. L. and ALBERT, R. (1999). Emergence of scaling in random networks. Science 286 509-512. MR2091634
4. BELKIN, M. (2003). Problems of learning on manifolds. Ph.D. thesis, Univ. Chicago.
5. BELKIN, M. and NIYOGI, P. (2003). Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput. 15 1373-1396. (Pubitemid 37049796)
6. BELKIN, M. andNIYOGI, P. (2008). Towards a theoretical foundation for Laplacian-based manifold methods. J. Comput. System Sci. 74 1289-1308. MR2460286
7. BHATIA, R. (1987). Perturbation Bounds for Matrix Eigenvalues. Longman, Harlow. MR0925418
8. BICKEL, P. J. and CHEN, A. (2009). A nonparametric view of network models and Newman-Girvan and other modularities. Proc. Nat. Acad. Sci. India Sect. A 106 21068.
9. BOCK, R. D. and HUSAIN, S. (1952). Factors of the Tele: A preliminary report. Sociometry 15 206-219.
10. BOUSQUET, O., CHAPELLE, O. and HEIN, M. (2004). Measure based regularization. Adv. Neural Inf. Process. Syst. 16.
11. BRANDES, U., DELLING, D., GAERTLER, M., GÖRKE, R., HOEFER, M., NIKOLOSKI, Z. and WAGNER, D. (2008). On modularity clustering. IEEE Transactions on Knowledge and Data Engineering 20 172-188.
12. BREIGER, R. L., BOORMAN, S. A. and ARABIE, P. (1975). An algorithm for clustering relational data with applications to social network analysis and comparison with multidimensional scaling. J. Math. Psych. 12 328-383.
13. CHAMPION, M. (1998). How many atoms make up the universe? Available at http://www.madsci.org/posts/archives/oct98/905633072.As.r.html.
14. CHOI, D., WOLFE, P. and AIROLDI, E. (2010). Stochastic blockmodels with growing number of classes. Preprint. Available at ArXiv:1011.4644.
15. COIFMAN, R., LAFON, S., LEE, A.,MAGGIONI, M., NADLER, B.,WARNER, F. and ZUCKER, S. (2005). Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps. Proc. Nat. Acad. Sci. India Sect. A 102 7426-7431. (Pubitemid 40740988)
16. CONDON, A. and KARP, R. M. (1999). Algorithms for graph partitioning on the planted partition model. In Randomization, Approximation, and Combinatorial Optimization (Berkeley, CA, 1999). Lecture Notes in Comput. Sci. 1671 221-232. Springer, Berlin. MR1775522
17. DONATH, W. E. and HOFFMAN, A. J. (1973). Lower bounds for the partitioning of graphs. IBM J. Res. Develop. 17 420-425. MR0329965
18. ERDÖS, P. and RÉNYI, A. (1959). On random graphs. Publ. Math. Debrecen 6 290-297.
19. FIEDLER, M. (1973). Algebraic connectivity of graphs. Czechoslovak Math. J. 23 298-305. MR0318007
20. FJÄLLSTRÖM, P. (1998). Algorithms for graph partitioning: A survey. Computer and Information Science 3. Available at http://www.ep.liu.se/ ea/cis/1998/010/.
21. FORTUNATO, S. (2010). Community detection in graphs. Phys. Rep. 486 75-174. MR2580414
22. FRANK, O. and STRAUSS, D. (1986). Markov graphs. J. Amer. Statist. Assoc. 81 832-842. MR0860518
23. FREEMAN, L. C. (2000). Visualizing social networks. Journal of Social Structure 1(1).
24. GINÉ, E. and KOLTCHINSKII, V. (2006). Empirical graph Laplacian approximation of Laplace- Beltrami operators: Large sample results. In High Dimensional Probability. Institute of Mathematical Statistics Lecture Notes-Monograph Series 51 238-259. IMS, Beachwood, OH. MR2387773
25. GIRVAN, M. and NEWMAN, M. (2002). Community structure in social and biological networks. Proc. Nat. Acad. Sci. India Sect. A 99 7821-7826. MR1908073 (Pubitemid 34650963)
26. 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.
27. HAGEN, L. and KAHNG, A. B. (1992). New spectral methods for ratio cut partitioning and clustering. IEEE Trans. Computer-Aided Design 11 1074-1085. (Pubitemid 23555831)
28. HANDCOCK, M. S., RAFTERY, A. E. and TANTRUM, J. M. (2007). Model-based clustering for social networks. J. Roy. Statist. Soc. Ser. A 170 301-354. MR2364300
29. HEIN, M. (2006). Uniform convergence of adaptive graph-based regularization. In Learning Theory. Lecture Notes in Comput. Sci. 4005 50-64. Springer, Berlin. MR2277918 (Pubitemid 44072185)
30. HEIN, M., AUDIBERT, J. Y. and VON LUXBURG, U. (2005). From graphs to manifolds-weak and strong pointwise consistency of graph Laplacians. In Learning Theory. Lecture Notes in Comput. Sci. 3559 470-485. Springer, Berlin. MR2203281
31. HENDRICKSON, B. and LELAND, R. (1995). An improved spectral graph partitioning algorithm for mapping parallel computations. SIAM J. Sci. Comput. 16 452-469. MR1317066
32. 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. MR1951262 (Pubitemid 36136568)
33. HOLLAND, P., LASKEY, K. B. and LEINHARDT, S. (1983). Stochastic blockmodels: Some first steps. Social Networks 5 109-137. MR0718088
34. HOLLAND, P.W. and LEINHARDT, S. (1981). An exponential family of probability distributions for directed graphs. J. Amer. Statist. Assoc. 76 33-50. MR0608176
35. KALLENBERG, O. (2005). Probabilistic Symmetries and Invariance Principles. Springer, New York. MR2161313
36. KLEIN, D. and RANDÍC , M. (1993). Resistance distance. J. Math. Chem. 12 81-95. MR1219566
37. KOREN, Y. (2005). Drawing graphs by eigenvectors: Theory and practice. Comput. Math. Appl. 49 1867-1888. MR2154691 (Pubitemid 41542517)
38. LAFON, S. S. (2004). Diffusion maps and geometric harmonics. Ph.D. thesis, Yale Univ.
39. LESKOVEC, J., LANG, K. J., DASGUPTA, A. and MAHONEY, M. W. (2008). Statistical properties of community structure in large social and information networks. In Proceeding of the 17th International Conference on World Wide Web 695-704. ACM, Beijing, China.
40. LIOTTA, G., ed. (2004). Graph Drawing. Lecture Notes in Comput. Sci. 2912. Springer, Berlin. MR2177577
41. MEILǍ, M. and SHI, J. (2001). A random walks view of spectral segmentation. In AI and Statistics (AISTATS) 2001.
42. NEWMAN, M. and GIRVAN, M. (2004). Finding and evaluating community structure in networks. Phys. Rev. E (3) 69 026113.
43. NOWICKI, K. and SNIJDERS, T. A. B. (2001). Estimation and prediction for stochastic blockstructures. J. Amer. Statist. Assoc. 96 1077-1087. MR1947255
44. POTHEN, A., SIMON, H. D. and LIOU, K. P. (1990). Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Anal. Appl. 11 430-452. MR1054210
45. SHI, J. and MALIK, J. (2000). Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 22 888-905.
46. SNIJDERS, T. A. B. andNOWICKI, K. (1997). Estimation and prediction for stochastic blockmodels for graphs with latent block structure. J. Classification 14 75-100. MR1449742
47. SPIELMAN, D. A. and TENG, S. H. (2007). Spectral partitioning works: Planar graphs and finite element meshes. Linear Algebra Appl. 421 284-305. MR2294342 (Pubitemid 46096985)
48. STEINHAUS, H. (1956). Sur la division des corp materiels en parties. Bull. Acad. Polon. Sci. Cl. III 4 801-804. MR0090073
49. STEWART, G.W. and SUN, J. (1990). Matrix Perturbation Theory. Academic Press, San Diego, CA.
50. TRAUD, A. L., KELSIC, E. D.,MUCHA, P. J. and PORTER, M. A. (2008). Community structure in online collegiate social networks. Preprint.
51. VAN DRIESSCHE, R. and ROOSE, D. (1995). An improved spectral bisection algorithm and its application to dynamic load balancing. Parallel Comput. 21 29-48. MR1314376
52. VAN DUIJN, M. A. J., SNIJDERS, T. A. B. and ZIJLSTRA, B. J. H. (2004). p2: A random effects model with covariates for directed graphs. Stat. Neerl. 58 234-254. MR2064846 (Pubitemid 38491680)
53. VON LUXBURG, U. (2007). A tutorial on spectral clustering. Stat. Comput. 17 395-416. MR2409803
54. VON LUXBURG, U., BELKIN, M. and BOUSQUET, O. (2008). Consistency of spectral clustering. Ann. Statist. 36 555-586. MR2396807
55. VU, V. (2011). Singular vectors under random perturbation. Random Structures Algorithms. DOI:10.1002/rsa.20367.
56. WASSERMAN, S. and FAUST, K. (1994). Social Network Analysis: Methods and Applications. Cambridge Univ. Press, Cambridge.
57. WATTS, D. J. and STROGATZ, S. H. (1998). Collective dynamics of 'small world' networks. Nature 393 440-442.