Advertisement

Local Lanczos Spectral Approximation for Community Detection

  • Pan Shi
  • Kun He
  • David Bindel
  • John E. Hopcroft
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10534)

Abstract

We propose a novel approach called the Local Lanczos Spectral Approximation (LLSA) for identifying all latent members of a local community from very few seed members. To reduce the computation complexity, we first apply a fast heat kernel diffusing to sample a comparatively small subgraph covering almost all possible community members around the seeds. Then starting from a normalized indicator vector of the seeds and by a few steps of Lanczos iteration on the sampled subgraph, a local eigenvector is gained for approximating the eigenvector of the transition matrix with the largest eigenvalue. Elements of this local eigenvector is a relaxed indicator for the affiliation probability of the corresponding nodes to the target community. We conduct extensive experiments on real-world datasets in various domains as well as synthetic datasets. Results show that the proposed method outperforms state-of-the-art local community detection algorithms. To the best of our knowledge, this is the first work to adapt the Lanczos method for local community detection, which is natural and potentially effective. Also, we did the first attempt of using heat kernel as a sampling method instead of detecting communities directly, which is proved empirically to be very efficient and effective.

Keywords

Community detection Heat kernel Local lanczos method 

Notes

Acknowledgments

The work is supported by NSFC (61772219, 61472147), US Army Research Office (W911NF-14-1-0477), and MSRA Collaborative Research (97354136).

References

  1. 1.
    Ahn, Y.Y., Bagrow, J.P., Lehmann, S.: Link communities reveal multiscale complexity in networks. Nature 466(7307), 761–764 (2010)CrossRefGoogle Scholar
  2. 2.
    Andersen, R., Chung, F., Lang, K.: Local graph partitioning using pagerank vectors. In: FOCS, pp. 475–486 (2006)Google Scholar
  3. 3.
    Andersen, R., Lang, K.J.: Communities from seed sets. In: WWW, pp. 223–232 (2006)Google Scholar
  4. 4.
    Andersen, R., Lang, K.J.: An algorithm for improving graph partitions. In: SODA, pp. 651–660 (2008)Google Scholar
  5. 5.
    Andersen, R., Peres, Y.: Finding sparse cuts locally using evolving sets. In: STOC, pp. 235–244 (2009)Google Scholar
  6. 6.
    Barnes, E.R.: An algorithm for partitioning the nodes of a graph. SIAM J. Algebraic Discrete Methods 3(4), 303–304 (1982)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bentbib, A.H., El Guide, M., Jbilou, K., Reichel, L.: A global Lanczos method for image restoration. J. Comput. Appl. Math. 300, 233–244 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chung, F.: The heat kernel as the pagerank of a graph. PNAS 104(50), 19735–19740 (2007)CrossRefGoogle Scholar
  9. 9.
    Chung, F., Simpson, O.: Solving linear systems with boundary conditions using heat kernel pagerank. In: Algorithms and Models for the Web Graph (WAW), pp. 203–219 (2013)Google Scholar
  10. 10.
    Chung, F.: A local graph partitioning algorithm using heat kernel pagerank. Internet Math. 6(3), 315–330 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chung, F.: Spectral Graph Theory, vol. 92. American Mathematical Society, Providence (1997)MATHGoogle Scholar
  12. 12.
    Coscia, M., Giannotti, F., Pedreschi, D.: A classification for community discovery methods in complex networks. Stastical Anal. Data Min. 4(5), 512–546 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Coscia, M., Rossetti, G., Giannotti, F., Pedreschi, D.: DEMON: a local-first discovery method for overlapping communities. In: KDD, pp. 615–623 (2012)Google Scholar
  14. 14.
    He, K., Sun, Y., Bindel, D., Hopcroft, J., Li, Y.: Detecting overlapping communities from local spectral subspaces. In: ICDM, pp. 769–774 (2015)Google Scholar
  15. 15.
    Kloster, K., Gleich, D.F.: Heat kernel based community detection. In: KDD, pp. 1386–1395 (2014)Google Scholar
  16. 16.
    Kloumann, I.M., Kleinberg, J.M.: Community membership identification from small seed sets. In: KDD, pp. 1366–1375 (2014)Google Scholar
  17. 17.
    Knight, P.A.: Fast rectangular matrix multiplication and QR decomposition. Linear Algebra Appl. 221, 69–81 (1995)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Komzsik, L.: The Lanczos Method: Evolution and Application, vol. 15. SIAM, Philadelphia (2003)CrossRefMATHGoogle Scholar
  19. 19.
    Lancichinetti, A., Fortunato, S.: Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities. Phys. Rev. E 80(1), 016118 (2009)CrossRefGoogle Scholar
  20. 20.
    Lancichinetti, A., Fortunato, S., Radicchi, F.: Benchmark graphs for testing community detection algorithms. Phys. Rev. E 78(4), 046110 (2008)CrossRefGoogle Scholar
  21. 21.
    Lanczos, C.: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Nat. Bur. Stan. 45, 255–282 (1950)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Li, Y., He, K., Bindel, D., Hopcroft, J.: Uncovering the small community structure in large networks. In: WWW, pp. 658–668 (2015)Google Scholar
  23. 23.
    Lütkepohl, H.: Handbook of Matrices, vol. 2. Wiley, Hoboken (1997)MATHGoogle Scholar
  24. 24.
    von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mahoney, M.W., Orecchia, L., Vishnoi, N.K.: A local spectral method for graphs: with applications to improving graph partitions and exploring data graphs locally. J. Mach. Learn. Res. 13(1), 2339–2365 (2012)MathSciNetMATHGoogle Scholar
  26. 26.
    Orecchia, L., Zhu, Z.A.: Flow-based algorithms for local graph clustering. In: SODA, pp. 1267–1286 (2014)Google Scholar
  27. 27.
    Paige, C.: The computation of eigenvalues and eigenvectors of very large sparse matrices. Ph.D. thesis, University of London (1971)Google Scholar
  28. 28.
    Palla, G., Derenyi, I., Farkas, I., Vicsek, T.: Uncovering the overlapping community structure of complex networks in nature and society. Nature 435(7043), 814–818 (2005)CrossRefGoogle Scholar
  29. 29.
    Parlett, B.N., Poole Jr., W.G.: A geometric theory for the QR, LU and power iterations. SIAM J. Numer. Anal. 10(2), 389–412 (1973)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22(8), 888–905 (2000)CrossRefGoogle Scholar
  31. 31.
    Spielman, D.A., Teng, S.: Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In: STOC, pp. 81–90 (2004)Google Scholar
  32. 32.
    Stewart, G.W.: On the early history of the singular value decomposition. SIAM Rev. 35(4), 551–566 (1993)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Wu, G., Xu, W., Leng, H.: Inexact and incremental bilinear Lanczos components algorithms for high dimensionality reduction and image reconstruction. Pattern Recogn. 48(1), 244–263 (2015)CrossRefMATHGoogle Scholar
  34. 34.
    Xie, J., Kelley, S., Szymanski, B.K.: Overlapping community detection in networks: the state-of-the-art and comparative study. ACM Comput. Surv. (CSUR) 45(4), 43 (2013)CrossRefMATHGoogle Scholar
  35. 35.
    Yang, J., Leskovec, J.: Defining and evaluating network communities based on ground-truth. In: ICDM, pp. 745–754 (2012)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Pan Shi
    • 1
  • Kun He
    • 1
    • 2
  • David Bindel
    • 2
  • John E. Hopcroft
    • 2
  1. 1.Huazhong University of Science and TechnologyWuhanChina
  2. 2.Cornell UniversityIthacaUSA

Personalised recommendations