An Energy Model for Visual Graph Clustering

  • Andreas Noack
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2912)


We introduce an energy model whose minimum energy drawings reveal the clusters of the drawn graph. Here a cluster is a set of nodes with many internal edges and few edges to nodes outside the set. The drawings of the best-known force and energy models do not clearly show clusters for graphs whose diameter is small relative to the number of nodes. We formally characterize the minimum energy drawings of our energy model. This characterization shows in what sense the drawings separate clusters, and how the distance of separated clusters to the other nodes can be interpreted.


Central Cluster Connected Graph Energy Model Visual Graph Internal Edge 
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.
    Aumann, Y., Rabani, Y.: An O(log k) approximate min-cut max-flow theorem and approximation algorithm. SIAM Journal on Computing 27(1), 291–301 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Battista, G.D., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, Englewood Cliffs (1999)zbMATHGoogle Scholar
  5. 5.
    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
  6. 6.
    Brandenburg, F.-J., Himsolt, M., Rohrer, C.: An experimental comparison of force-directed and randomized graph drawing algorithms. In: Brandenburg, F.J. (ed.) GD 1995. LNCS, vol. 1027, pp. 76–87. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  7. 7.
    Brandes, U.: Drawing on physical analogies. In: Kaufmann, M., Wagner, D. (eds.) Drawing Graphs. LNCS, vol. 2025, pp. 71–86. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Brandes, U., Cornelsen, S.: Visual ranking of link structures. In: Dehne, F., Sack, J.-R., Tamassia, R. (eds.) WADS 2001. LNCS, vol. 2125, pp. 222–233. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. 9.
    Brockenauer, R., Cornelsen, S.: Drawing clusters and hierarchies. In: Kaufmann, M., Wagner, D. (eds.) Drawing Graphs. LNCS, vol. 2025, p. 193. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. 10.
    Davidson, R., Harel, D.: Drawing graphs nicely using simulated annealing. ACM Transactions on Graphics 15(4), 301–331 (1996)CrossRefGoogle Scholar
  11. 11.
    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
  12. 12.
    Eades, P.: A heuristic for graph drawing. Congressus Numerantium 42, 149–160 (1984)MathSciNetGoogle Scholar
  13. 13.
    Eades, P., Huang, M.L.: Navigating clustered graphs using force-directed methods. Journal of Graph Algorithms and Applications 4(3), 157–181 (2000)zbMATHGoogle Scholar
  14. 14.
    Fruchterman, T.M.J., Reingold, E.M.: Graph drawing by force-directed placement. Software – Practice and Experience 21(11), 1129–1164 (1991)CrossRefGoogle Scholar
  15. 15.
    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
  16. 16.
    Hall, K.M.: An r-dimensional quadratic placement algorithm. Management Science 17(3), 219–229 (1970)zbMATHCrossRefGoogle Scholar
  17. 17.
    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
  18. 18.
    Kamada, T., Kawai, S.: An algorithm for drawing general undirected graphs. Information Processing Letters 31(1), 7–15 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Koren, Y., Carmel, L., Harel, D.: Ace: A fast multiscale eigenvector computation for drawing huge graphs. In: Proc. IEEE Symposium on Information Visualization (InfoVis 2002), pp. 137–144 (2002)Google Scholar
  20. 20.
    Kruskal, J.B., Wish, M.: Multidimensional Scaling. Sage Publications, Thousand Oaks (1978)Google Scholar
  21. 21.
    Lewerentz, C., Noack, A.: CrocoCosmos – 3D visualization of large object-oriented programs. In: Graph Drawing Software, pp. 279–297. Springer, Heidelberg (2003)Google Scholar
  22. 22.
    Linial, N., London, E., Rabinovich, Y.: The geometry of graphs and some of its algorithmic applications. Combinatorica 15(2), 215–245 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    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 Understanding (IWPC 1998), pp. 45–52 (1998)Google Scholar
  24. 24.
    Noack, A.: Energy models for drawing clustered small-world graphs. Technical Report 07/03, Institute of Computer Science, Brandenburg University of Technology at Cottbus (2003)Google Scholar
  25. 25.
    Pothen, A.: Graph partitioning algorithms with applications to scientific computing. In: Keyes, D.E., Sameh, A., Venkatakrishnan, V. (eds.) Parallel Numerical Algorithms, pp. 323–368. Kluwer, Dordrecht (1997)Google Scholar
  26. 26.
    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
  27. 27.
    Strogatz, S.H.: Exploring complex networks. Nature 410, 268–276 (2001)CrossRefGoogle Scholar
  28. 28.
    Tunkelang, D.: A Numerical Optimization Approach to General Graph Drawing. PhD thesis, School of Computer Science, Carnegie Mellon University (1999)Google Scholar
  29. 29.
    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
  30. 30.
    Wang, X., Miyamoto, I.: Generating customized layouts. In: Brandenburg, F.J. (ed.) GD 1995. LNCS, vol. 1027, pp. 504–515. Springer, Heidelberg (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Andreas Noack
    • 1
  1. 1.Institute of Computer Science BrandenburgTechnical University at CottbusCottbusGermany

Personalised recommendations