Advertisement

Consistent Recovery of Communities from Sparse Multi-relational Networks: A Scalable Algorithm with Optimal Recovery Conditions

  • Sharmodeep Bhattacharyya
  • Shirshendu ChatterjeeEmail author
Conference paper
  • 77 Downloads
Part of the Springer Proceedings in Complexity book series (SPCOM)

Abstract

Multi-layer and multiplex networks show up frequently in many recent network datasets. We consider the problem of identifying the common community membership structure of a finite sequence of networks, called multi-relational networks, which can be considered a particular case of multiplex and multi-layer networks. We propose two scalable spectral methods for identifying communities within a finite sequence of networks. We provide theoretical results to quantify the performance of the proposed methods when individual networks are generated from either the stochastic block model or the degree-corrected block model. The methods are guaranteed to recover communities consistently when either the number of networks goes to infinity arbitrarily slowly, or the expected degree of a typical node goes to infinity arbitrarily slowly, even if all the individual networks have fixed size and are sparse. This condition on the parameters of the network models mentioned above is both sufficient for consistent community recovery using our methods and also necessary to have any consistent community detection procedure. We also give some simulation results to demonstrate the efficacy of the proposed methods.

Keywords

Spectral clustering Community detection Multi-relational networks Multi-layer networks Stochastic block model Degree-corrected block model 

Supplementary material

494520_1_En_9_MOESM1_ESM.pdf (379 kb)
Supplementary material 1 (pdf 379 KB)

References

  1. 1.
    Bhattacharyya, S., Bickel, P.J.: Community detection in networks using graph distance. arXiv preprint arXiv:1401.3915 (2014)
  2. 2.
    Chen, P.Y., Hero, A.O.: Multilayer spectral graph clustering via convex layer aggregation: theory and algorithms. IEEE Trans. Signal Inf. Process. Netw. 3(3), 553–567 (2017)CrossRefGoogle Scholar
  3. 3.
    Dong, X., Frossard, P., Vandergheynst, P., Nefedov, N.: Clustering with multi-layer graphs: a spectral perspective. IEEE Trans. Signal Process. 60(11), 5820–5831 (2012)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Drineas, P., Kannan, R., Mahoney, M.W.: Fast Monte Carlo algorithms for matrices II: computing a low-rank approximation to a matrix. SIAM J. Comput. 36(1), 158–183 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Feige, U., Ofek, E.: Spectral techniques applied to sparse random graphs. Random Struct. Algorithms 27(2), 251–275 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Feldman, D., Monemizadeh, M., Sohler, C.: A PTAS for k-means clustering based on weak coresets. In: Proceedings of the Twenty-Third Annual Symposium on Computational Geometry, pp. 11–18. ACM (2007)Google Scholar
  7. 7.
    Fiedler, M.: Algebraic connectivity of graphs. Czechoslovak Math. J. 23(98), 298–305 (1973)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Fortunato, S.: Community detection in graphs. Phys. Rep. 486(3–5), 75–174 (2010)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Halko, N., Martinsson, P.G., Tropp, J.A.: Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53(2), 217–288 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Han, Q., Xu, K., Airoldi, E.: Consistent estimation of dynamic and multi-layer block models. In: International Conference on Machine Learning, pp. 1511–1520 (2015)Google Scholar
  11. 11.
    Holland, P.W., Laskey, K.B., Leinhardt, S.: Stochastic blockmodels: first steps. Soc. Netw. 5(2), 109–137 (1983)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Jin, J., et al.: Fast community detection by score. Ann. Stat. 43(1), 57–89 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Karrer, B., Newman, M.E.: Stochastic blockmodels and community structure in networks. Phys. Rev. E 83(1), 016107 (2011)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Kumar, A., Rai, P., Daumé III, H.: Co-regularized spectral clustering with multiple kernels (2010)Google Scholar
  15. 15.
    Kumar, A., Sabharwal, Y., Sen, S.: A simple linear time (1+ \(\varepsilon \))-approximation algorithm for k-means clustering in any dimensions. In: Annual Symposium on Foundations of Computer Science, pp. 454–462 (2004)Google Scholar
  16. 16.
    Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, vol. 6. SIAM, Philadelphia (1998)Google Scholar
  17. 17.
    Lei, J., Rinaldo, A., et al.: Consistency of spectral clustering in stochastic block models. Ann. Stat. 43(1), 215–237 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lu, L., Peng, X.: Spectra of edge-independent random graphs. Electron. J. Comb. 20(4), P27 (2013)MathSciNetzbMATHGoogle Scholar
  19. 19.
    von Luxburg, U., Belkin, M., Bousquet, O.: Consistency of spectral clustering. Ann. Statist. 36(2), 555–586 (2008).  https://doi.org/10.1214/009053607000000640MathSciNetCrossRefGoogle Scholar
  20. 20.
    Matias, C., Miele, V.: Statistical clustering of temporal networks through a dynamic stochastic block model. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 79(4), 1119–1141 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ng, A.Y., Jordan, M.I., Weiss, Y., et al.: On spectral clustering: analysis and an algorithm. Adv. Neural Inf. Process. Syst. 2, 849–856 (2002)Google Scholar
  22. 22.
    Paul, S., Chen, Y.: Spectral and matrix factorization methods for consistent community detection in multi-layer networks. arXiv preprint arXiv:1704.07353 (2017)
  23. 23.
    Paul, S., Chen, Y., et al.: Consistent community detection in multi-relational data through restricted multi-layer stochastic blockmodel. Electron. J. Stat. 10(2), 3807–3870 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Pensky, M., Zhang, T., et al.: Spectral clustering in the dynamic stochastic block model. Electron. J. Stat. 13(1), 678–709 (2019)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rohe, K., Chatterjee, S., Yu, B.: Spectral clustering and the high-dimensional stochastic blockmodel. Ann. Statist. 39(4), 1878–1915 (2011).  https://doi.org/10.1214/11-AOS887MathSciNetCrossRefGoogle Scholar
  26. 26.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22(8), 888–905 (2000)CrossRefGoogle Scholar
  27. 27.
    Sussman, D.L., Tang, M., Fishkind, D.E., Priebe, C.E.: A consistent adjacency spectral embedding for stochastic blockmodel graphs. J. Am. Stat. Assoc. 107(499), 1119–1128 (2012)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Tang, W., Lu, Z., Dhillon, I.S.: Clustering with multiple graphs. In: Ninth IEEE International Conference on Data Mining 2009. ICDM 2009, pp. 1016–1021. IEEE (2009)Google Scholar
  29. 29.
    Xu, K.S., Hero, A.O.: Dynamic stochastic blockmodels for time-evolving social networks. IEEE J. Sel. Top. Signal Process. 8(4), 552–562 (2014)ADSCrossRefGoogle Scholar
  30. 30.
    Young, S.J., Scheinerman, E.R.: Random dot product graph models for social networks. In: International Workshop on Algorithms and Models for the Web-Graph, pp. 138–149. Springer, Heidelberg (2007)Google Scholar
  31. 31.
    Zhang, A.Y., Zhou, H.H., et al.: Minimax rates of community detection in stochastic block models. Ann. Stat. 44(5), 2252–2280 (2016)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Zhang, X., Moore, C., Newman, M.E.: Random graph models for dynamic networks. Eur. Phys. J. B 90(10), 200 (2017)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Sharmodeep Bhattacharyya
    • 1
  • Shirshendu Chatterjee
    • 2
    Email author
  1. 1.Oregon State UniversityCorvallisUSA
  2. 2.City University of New YorkNew YorkUSA

Personalised recommendations