Local Clustering of Large Graphs by Approximate Fiedler Vectors

  • Pekka Orponen
  • Satu Elisa Schaeffer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3503)


We address the problem of determining the natural neighbourhood of a given node i in a large nonunifom network G in a way that uses only local computations, i.e. without recourse to the full adjacency matrix of G. We view the problem as that of computing potential values in a diffusive system, where node i is fixed at zero potential, and the potentials at the other nodes are then induced by the adjacency relation of G. This point of view leads to a constrained spectral clustering approach. We observe that a gradient method for computing the respective Fiedler vector values at each node can be implemented in a local manner, leading to our eventual algorithm. The algorithm is evaluated experimentally using three types of nonuniform networks: randomised “caveman graphs”, a scientific collaboration network, and a small social interaction network.


Source Node Local Cluster Large Graph Simple Random Walk Global Cluster 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brandes, U., Gaertler, M., Wagner, D.: Experiments on graph clustering algorithms. In: Di Battista, G., Zwick, U. (eds.) ESA 2003, vol. 2832, pp. 568–579. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Brémaud, P.: Markov Chains, Gibbs Fields, Monte Carlo Simulation, and Queues. Springer, New York (1999)zbMATHGoogle Scholar
  3. 3.
    Chung, F.R.K.: Spectral Graph Theory, American Mathematical Society, Providence, RI (1997)Google Scholar
  4. 4.
    Chung, F.R.K., Ellis, R.B.: A chip-firing game and Dirichlet eigenvalues. Discrete Mathematics 257, 341–355 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Doyle, P.G., Snell, J.L.: Random Walks and Electric Networks. In: Mathematical Association of America, Washington, DC (1984)Google Scholar
  6. 6.
    Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification. Wiley, New York (2001)zbMATHGoogle Scholar
  7. 7.
    Fiedler, M.: Algebraic connectivity of graphs. Czechoslovak Mathematical Journal 23, 298–305 (1973)MathSciNetGoogle Scholar
  8. 8.
    Fiedler, M.: A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory. Czechoslovak Mathematical Journal 25, 619–633 (1975)MathSciNetGoogle Scholar
  9. 9.
    Flake, G.W., Lawrence, S., Giles, C.L., Coetzee, F.M.: Self-organization and identification of Web communities. IEEE Computer 35(3), 66–71 (2002)Google Scholar
  10. 10.
    Gkantsidis, C., Mihail, M., Zegura, E.: Spectral analysis of Internet topologies. In: Proceedings of the 22nd Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM 2003), pp. 364–374. IEEE, New York (2003)Google Scholar
  11. 11.
    Guattery, S., Miller, G.L.: On the quality of spectral separators. SIAM Journal on Matrix Analysis and Applications 19(3), 701–719 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    He, X., Zha, H., Ding, C.H.Q., Simon, H.: Web document clustering using hyperlink structures. Computational Statistics & Data Analysis 41(1), 19–45 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hopcroft, J., Khan, O., Kulis, B., Selman, B.: Natural communities in large linked networks. In: Proceedings of the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 541–546. ACM, New York (2003)CrossRefGoogle Scholar
  14. 14.
    Kannan, R., Vempala, S., Vetta, A.: On clusterings: Good, bad and spectral. Journal of the ACM 51(3), 497–515 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Kempe, D., McSherry, F.: A decentralized algorithm for spectral analysis. In: Proceedings of the 36th ACM Symposium on Theory of Computing (STOC 2004). ACM, New York (2004)Google Scholar
  16. 16.
    Newman, M.E.J.: The structure and function of complex networks. SIAM Review 45, 167–256 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Newman, M.E.J.: Fast algorithm for detecting community structure in networks. Physical Review E 69, 066113 (2004)Google Scholar
  18. 18.
    Newman, M.E.J., Girvan, M.: Finding and evaluating community structure in networks. Physical Review E 69, 026113 (2004)Google Scholar
  19. 19.
    Pothen, A., Simon, H.D., Liou, K.P.: Partitioning sparse matrices with eigenvectors of graphs. SIAM Journal of Matrix Analysis and Applications 11, 430–452 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Schaeffer, S.E.: Stochastic local clustering for massive graphs. In: Ho, T.-B., Cheung, D., Liu, H. (eds.) PAKDD 2005. LNCS, vol. 3518, pp. 354–360. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  21. 21.
    Spielman, D.A., Teng, S.-H.: Spectral partitioning works: planar graphs and finite element meshes. In: Proceedings of the 37th IEEE Symposium on Foundations of Computing (FOCS 1996), pp. 96–105. IEEE Computer Society, Los Alamitos (1996)Google Scholar
  22. 22.
    Virtanen, S.E.: Clustering the Chilean Web. In: Proceedings of the First Latin American Web Congress, pp. 229–231. IEEE Computer Society, Los Alamitos (2003)Google Scholar
  23. 23.
    Virtanen, S.E.: Properties of nonuniform random graph models. Research Report A77, Helsinki University of Technology, Laboratory for Theoretical Computer Science, Espoo, Finland (May 2003),
  24. 24.
    Watts, D.J.: Small Worlds: The Dynamics of Networks between Ordeer and Randomness. Princeton University Press, Princeton (1999)Google Scholar
  25. 25.
    Wu, F., Huberman, B.A.: Finding communities in linear time: a physics approach. The European Physics Journal B 38, 331–338 (2004)CrossRefGoogle Scholar
  26. 26.
    Zachary, W.W.: An information flow model for conflict and fission in small groups. Journal of Anthropological Research 33, 452–473 (1977)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Pekka Orponen
    • 1
  • Satu Elisa Schaeffer
    • 1
  1. 1.Laboratory for Theoretical Computer ScienceTKK Helsinki University of TechnologyFinland

Personalised recommendations