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Bipartite Communities via Spectral Partitioning

  • Kelly B. Yancey
  • Matthew P. Yancey
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11346)

Abstract

In this paper we are interested in finding communities with bipartite structure. A bipartite community is a pair of disjoint vertex sets S, \(S'\) such that the number of edges with one endpoint in S and the other endpoint in \(S'\) is “significantly more than expected.” This additional structure is natural to some applications of community detection. In fact, using other terminology, they have already been used to study correlation networks, social networks, and two distinct biological networks.

In 2012 two groups independently ((1) Lee, Oveis Gharan, and Trevisan and (2) Louis, Raghavendra, Tetali, and Vempala) used higher eigenvalues of the normalized Laplacian to find an approximate solution to the k-sparse-cuts problem. In 2015 Liu generalized spectral methods for finding k communities to find k bipartite communities. Our approach improves the bounds on bipartite conductance (measure of strength of a bipartite community) found by Liu and also implies improvements to the original spectral methods by Lee et al. and Louis et al. We also highlight experimental results found when applying our algorithm to three distinct real-world networks.

Keywords

Community detection Spectral graph theory Network analysis 

References

  1. 1.
    Arora, S., Barak, B., Steurer, D.: Subexponential algorithms for unique games and related problems. J. ACM 62(5), 25 (2015).  https://doi.org/10.1145/2775105. Art. 42MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bauer, F., Jost, J.: Bipartite and neighborhood graphs and the spectrum of the normalized graph Laplace operator. Commun. Anal. Geom. 21(4), 787–845 (2013).  https://doi.org/10.4310/CAG.2013.v21.n4.a2MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bellay, J., et al.: Putting genetic interactions in context through a global modular decomposition. Genome Res. 21(8), 1375–1387 (2011).  https://doi.org/10.1101/gr.117176.110CrossRefGoogle Scholar
  4. 4.
    Buluç, A., Meyerhenke, H., Safro, I., Sanders, P., Schulz, C.: Recent advances in graph partitioning. In: Kliemann, L., Sanders, P. (eds.) Algorithm Engineering. LNCS, vol. 9220, pp. 117–158. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-49487-6_4CrossRefGoogle Scholar
  5. 5.
    Charikar, M., Chekuri, C., Goel, A., Guha, S., Plotkin, S.: Approximating a finite metric by a small number of tree metrics. In: Proceedings of the 39th Annual Symposium on Foundations of Computer Science. FOCS 1998, p. 379. IEEE Computer Society, Washington, DC (1998). http://dl.acm.org/citation.cfm?id=795664.796406
  6. 6.
    Chung, F.: Four Cheeger-type inequalities for graph partitioning algorithms. In: Proceedings of ICCM, pp. 751–772 (2007)Google Scholar
  7. 7.
    Gallier, J.: Spectral theory of unsigned and signed graphs. Applications to graph clustering: a survey. ArXiv e-prints, January 2016Google Scholar
  8. 8.
    Gharan, S.O., Trevisan, L.: Partitioning into expanders. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1256–1266. ACM, New York (2014).  https://doi.org/10.1137/1.9781611973402.93Google Scholar
  9. 9.
    Kleinberg, J.M.: Authoritative sources in a hyperlinked environment. J. ACM 46(5), 604–632 (1999).  https://doi.org/10.1145/324133.324140MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kolev, P., Mehlhorn, K.: A note on spectral clustering. In: 24th Annual European Symposium on Algorithms. Leibniz International Proceedings in Informatics, LIPIcs, vol. 57, p. 14, Art. No. 57. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2016)Google Scholar
  11. 11.
    Lee, J.R., Oveis Gharan, S., Trevisan, L.: Multi-way spectral partitioning and higher-order Cheeger inequalities. In: Proceedings of the 2012 ACM Symposium on Theory of Computing, STOC 2012, pp. 1117–1130. ACM, New York (2012).  https://doi.org/10.1145/2213977.2214078
  12. 12.
    Li, A., Peng, P.: Detecting and characterizing small dense bipartite-like subgraphs by the bipartiteness ratio measure. In: Cai, L., Cheng, S.-W., Lam, T.-W. (eds.) ISAAC 2013. LNCS, vol. 8283, pp. 655–665. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-45030-3_61CrossRefzbMATHGoogle Scholar
  13. 13.
    Li, J., Liu, G., Li, H., Wong, L.: Maximal biclique subgraphs and closed pattern pairs of the adjacency matrix: a one-to-one correspondence and mining algorithms. IEEE Trans. Knowl. Data Eng. 19(12), 1625–1637 (2007).  https://doi.org/10.1109/TKDE.2007.190660CrossRefGoogle Scholar
  14. 14.
    Liu, F.M.A.S.: Cheeger constants, structural balance, and spectral clustering analysis for signed graphs. Max Planck Institute for Mathematics in the Sciences (2014, Preprint). http://www.mis.mpg.de/de/publications/preprints/2014/prepr2014-111.html
  15. 15.
    Liu, S.: Multi-way dual Cheeger constants and spectral bounds of graphs. Adv. Math. 268, 306–338 (2015).  https://doi.org/10.1016/j.aim.2014.09.023MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lo, D., Surian, D., Prasetyo, P.K., Zhang, K., Lim, E.P.: Mining direct antagonistic communities in signed social networks. Inf. Process. Manag. 49(4), 773–791 (2013).  https://doi.org/10.1016/j.ipm.2012.12.009CrossRefGoogle Scholar
  17. 17.
    Louis, A., Raghavendra, P., Tetali, P., Vempala, S.: Many sparse cuts via higher eigenvalues. In: Proceedings of the 2012 ACM Symposium on Theory of Computing, STOC 2012, pp. 1131–1140. ACM, New York (2012).  https://doi.org/10.1145/2213977.2214079
  18. 18.
    Nascimento, M.C.V., de Carvalho, A.C.P.L.F.: Spectral methods for graph clustering – a survey. Eur. J. Oper. Res. 211(2), 221–231 (2011).  https://doi.org/10.1016/j.ejor.2010.08.012MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Peng, R., Sun, H., Zanetti, L.: Partitioning well-clustered graphs: spectral clustering works!. SIAM J. Comput. 46(2), 710–743 (2017).  https://doi.org/10.1137/15M1047209MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rohe, K., Chatterjee, S., Yu, B.: Spectral clustering and the high-dimensional stochastic blockmodel. Ann. Stat. 39(4), 1878–1915 (2011).  https://doi.org/10.1214/11-AOS887MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Trevisan, L.: Max cut and the smallest eigenvalue. SIAM J. Comput. 41(6), 1769–1786 (2012).  https://doi.org/10.1137/090773714MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Verma, D., Meila, M.: A comparison of spectral clustering algorithms. Technical report, University of Washington CSE (2003)Google Scholar
  23. 23.
    Wolf, R.D.: A brief introduction to Fourier analysis on the Boolean cube. Theory of Computing Library Graduate Surveys (2008)Google Scholar
  24. 24.
    Yancey, K., Yancey, M.: Bipartite Communities. ArXiv e-prints, December 2014Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Institute for Defense Analyses - Center for Computing SciencesUniversity of MarylandBowieUSA
  2. 2.Institute for Defense Analyses - Center for Computing SciencesUniversity of MarylandBowieUSA

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