Energy-Based Clustering of Graphs with Nonuniform Degrees

  • Andreas Noack
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3843)


Widely varying node degrees occur in software dependency graphs, hyperlink structures, social networks, and many other real-world graphs. Finding dense subgraphs in such graphs is of great practical interest, as these clusters may correspond to cohesive software modules, semantically related documents, and groups of friends or collaborators. Many existing clustering criteria and energy models are biased towards clustering together nodes with high degrees. In this paper, we introduce a clustering criterion based on normalizing cuts with edge numbers (instead of node numbers), and a corresponding energy model based on edge repulsion (instead of node repulsion) that reveal clusters without this bias.


Energy Model Large Graph Cluster Criterion Graph Cluster Node Versus 
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.


  1. 1.
    Albert, R., Barabási, A.-L.: Statistical mechanics of complex networks. Reviews of Modern Physics 74(1), 47–97 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Alpert, C.J., Kahng, A.B.: Recent directions in netlist partitioning: A survey. Integration, the VLSI Journal 19(1-2), 1–81 (1995)zbMATHCrossRefGoogle Scholar
  3. 3.
    Barnes, J., Hut, P.: A hierarchical O(N log N) force-calculation algorithm. Nature 324, 446–449 (1986)CrossRefGoogle Scholar
  4. 4.
    Blythe, J., McGrath, C., Krackhardt, D.: The effect of graph layout on inference from social network data. In: Brandenburg, F.J. (ed.) GD 1995. LNCS, vol. 1027, pp. 40–51. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  5. 5.
    Davidson, R., Harel, D.: Drawing graphs nicely using simulated annealing. ACM Transactions on Graphics 15(4), 301–331 (1996)CrossRefGoogle Scholar
  6. 6.
    Dengler, E., Cowan, W.: Human perception of laid-out graphs. In: Whitesides, S.H. (ed.) GD 1998. LNCS, vol. 1547, pp. 441–443. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  7. 7.
    Eades, P.: A heuristic for graph drawing. Congressus Numerantium 42, 149–160 (1984)MathSciNetGoogle Scholar
  8. 8.
    Fruchterman, T.M.J., Reingold, E.M.: Graph drawing by force-directed placement. Software – Practice and Experience 21(11), 1129–1164 (1991)CrossRefGoogle Scholar
  9. 9.
    Gajer, P., Goodrich, M.T., Kobourov, S.G.: A multi-dimensional approach to force-directed layouts of large graphs. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 211–221. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. 10.
    Hachul, S., Jünger, M.: Drawing large graphs with a potential-field-based multilevel algorithm. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 285–295. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Hall, K.M.: An r-dimensional quadratic placement algorithm. Management Science 17(3), 219–229 (1970)zbMATHCrossRefGoogle Scholar
  12. 12.
    Harel, D., Koren, Y.: A fast multi-scale method for drawing large graphs. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 183–196. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  13. 13.
    Kamada, T., Kawai, S.: An algorithm for drawing general undirected graphs. Information Processing Letters 31(1), 7–15 (1989)zbMATHCrossRefMathSciNetGoogle 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.
    Leighton, T., Rao, S.: An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms. In: Proc. 29th Annual Symposium on Foundations of Computer Science (FOCS 1988), pp. 422–431. IEEE, Los Alamitos (1988)Google Scholar
  16. 16.
    Mancoridis, S., Mitchell, B.S., Rorres, C., Chen, Y., Gansner, E.R.: Using automatic clustering to produce high-level system organizations of source code. In: Proc. 6th IEEE International Workshop on Program Comprehension (IWPC 1998), pp. 45–52. IEEE, Los Alamitos (1998)Google Scholar
  17. 17.
    Noack, A.: An energy model for visual graph clustering. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 425–436. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. 18.
    Noack, A., Lewerentz, C.: A space of layout styles for hierarchical graph models of software systems. In: Proc. 2nd ACM Symposium on Software Visualization (SoftVis 2005), pp. 155–164. ACM, New York (2005)CrossRefGoogle Scholar
  19. 19.
    Quigley, A.J., Eades, P.: FADE: Graph drawing, clustering, and visual abstraction. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 197–210. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  20. 20.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Transaction on Pattern Analysis and Machine Intelligence 22(8), 888–905 (2000)CrossRefGoogle Scholar
  21. 21.
    Walshaw, C.: A multilevel algorithm for force-directed graph drawing. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 171–182. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  22. 22.
    Wu, Z., Leahy, R.: An optimal graph theoretic approach to data clustering: Theory and its application to image segmentation. IEEE Transaction on Pattern Analysis and Machine Intelligence 15(11), 1101–1113 (1993)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Andreas Noack
    • 1
  1. 1.Institute of Computer ScienceBrandenburg University of Technology at CottbusCottbusGermany

Personalised recommendations