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An Efficient Generator for Clustered Dynamic Random Networks

  • Robert Görke
  • Roland Kluge
  • Andrea Schumm
  • Christian Staudt
  • Dorothea Wagner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7659)

Abstract

A planted partition graph is an Erdős-Rényi type random graph, where, based on a given partition of the vertex set, vertices in the same part are linked with a higher probability than vertices in different parts. Graphs of this type are frequently used to evaluate graph clustering algorithms, i.e., algorithms that seek to partition the vertex set of a graph into densely connected clusters. We propose a self-evident modification of this model to generate sequences of random graphs that are obtained by atomic updates, i.e., the deletion or insertion of an edge or vertex. The random process follows a dynamically changing ground-truth clustering that can be used to evaluate dynamic graph clustering algorithms. We give a theoretical justification of our model and show how the corresponding random process can be implemented efficiently.

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References

  1. 1.
    Fortunato, S.: Community detection in graphs. Physics Reports 486(3-5), 75–174 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Schaeffer, S.E.: Graph Clustering. Computer Science Review 1(1), 27–64 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bollobás, B.: Random Graphs. Cambridge University Press (2001)Google Scholar
  4. 4.
    Gilbert, H.: Random Graphs. The Annals of Mathematical Statistics 30(4), 1141–1144 (1959)zbMATHCrossRefGoogle Scholar
  5. 5.
    Condon, A., Karp, R.M.: Algorithms for Graph Partitioning on the Planted Partition Model. Randoms Structures and Algorithms 18(2), 116–140 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Brandes, U., Gaertler, M., Wagner, D.: Experiments on Graph Clustering Algorithms. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 568–579. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Gaertler, M., Görke, R., Wagner, D.: Significance-Driven Graph Clustering. In: Kao, M.-Y., Li, X.-Y. (eds.) AAIM 2007. LNCS, vol. 4508, pp. 11–26. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Girvan, M., Newman, M.E.J.: Community structure in social and biological networks. Proceedings of the National Academy of Science of the United States of America 99(12), 7821–7826 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Watts, D.J.: Small worlds: The dynamics of networks between order and randomness. Princeton University Press (1999)Google Scholar
  10. 10.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of ’small-world’ networks. Nature 393(6684), 440–442 (1998)CrossRefGoogle Scholar
  11. 11.
    Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Leskovec, J., Kleinberg, J.M., Faloutsos, C.: Graphs Over Time: Densification Laws, Shrinking Diameters and Possible Explanations. In: Proceedings of the 11th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 177–187. ACM Press (2005)Google Scholar
  13. 13.
    Vázquez, A.: Growing network with local rules: Preferential attachment, clustering hierarchy, and degree correlations. Physical Review E 67, 056104 (2003)Google Scholar
  14. 14.
    Bagrow, J.: Evaluating local community methods in networks. Journal of Statistical Mechanics: Theory and Experiment, P05001 (2008), doi:10.1088/1742-5468/2008/05/P05001Google Scholar
  15. 15.
    Lancichinetti, A., Fortunato, S.: Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities. Physical Review E 80(1), 016118 (2009)Google Scholar
  16. 16.
    Fan, Y., Li, M., Zhang, P., Wu, J., Di, Z.: Accuracy and precision of methods for community identification in weighted networks. Physica A 377(1), 363–372 (2007)CrossRefGoogle Scholar
  17. 17.
    Guimerà, R., Sales-Pardo, M., Amaral, L.A.N.: Module identification in bipartite and directed networks. Physical Review E 76, 036102 (2007)Google Scholar
  18. 18.
    Zhou, H.: Network landscape from a Brownian particle’s perspective. Physical Review E 67, 041908 (2003)Google Scholar
  19. 19.
    Sawardecker, E.N., Sales-Pardo, M., Amaral, L.A.N.: Detection of node group membership in networks with group overlap. The European Physical Journal B 67, 277–284 (2009)Google Scholar
  20. 20.
    Aldecoa, R., Marín, I.: Closed benchmarks for network community structure characterization. Physical Review E 85, 026109 (2012)Google Scholar
  21. 21.
    Brandes, U., Mader, M.: A Quantitative Comparison of Stress-Minimization Approaches for Offline Dynamic Graph Drawing. In: van Kreveld, M., Speckmann, B. (eds.) GD 2011. LNCS, vol. 7034, pp. 99–110. Springer, Heidelberg (2011)Google Scholar
  22. 22.
    Robins, G., Pattison, P., Kalish, Y., Lusher, D.: An introduction to exponential random graph (p*) models for social networks. Social Networks 29(2), 173–191 (2007)CrossRefGoogle Scholar
  23. 23.
    Snijders, T.A.: The Statistical Evaluation of Social Network Dynamics. Sociological Methodology 31(1), 361–395 (2001)CrossRefGoogle Scholar
  24. 24.
    Clementi, A.E.F., Macci, C., Monti, A., Pasquale, F., Silvestri, R.: Flooding time in edge-Markovian dynamic graphs. SIAM Journal on Discrete Mathematics 24(4), 1694–1712 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Baumann, H., Crescenzi, P., Fraigniaud, P.: Parsimonious flooding in dynamic graphs. In: Proceedings of the 28th ACM Symposium on Principles of Distributed Computing, pp. 260–269. ACM Press (2009)Google Scholar
  26. 26.
    Görke, R., Staudt, C.: A Generator for Dynamic Clustered Random Graphs. Technical report, Informatik, Uni Karlsruhe, TR 2009-7 (2009)Google Scholar
  27. 27.
    Görke, R.: An Algorithmic Walk from Static to Dynamic Graph Clustering. PhD thesis, Fakultät für Informatik (February 2010)Google Scholar
  28. 28.
    Görke, R., Maillard, P., Staudt, C., Wagner, D.: Modularity-Driven Clustering of Dynamic Graphs. In: Festa, P. (ed.) SEA 2010. LNCS, vol. 6049, pp. 436–448. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  29. 29.
    Görke, R., Kluge, R., Schumm, A., Staudt, C., Wagner, D.: An Efficient Generator for Clustered Dynamic Random Networks. Technical report, Karlsruhe Reports in Informatics 2012, 17 (2012)Google Scholar
  30. 30.
    Behrends, E.: Introduction to Markov Chains With Special Emphasis on Rapid Mixing. Friedrick Vieweg & Son (October 2002)Google Scholar
  31. 31.
    Batagelj, V., Brandes, U.: Efficient Generation of Large Random Networks. Physical Review E 036113 (2005)Google Scholar
  32. 32.
    Fisher, R.A., Yates, F.: Statistical Tables for Biological, Agricultural and Medical Research. Oliver and Boyd, London (1948)zbMATHGoogle Scholar
  33. 33.
    Fan, C.T., Muller, M.E., Rezucha, I.: Development of Sampling Plans by Using Sequential (Item by Item) Selection Techniques and Digital-Computers. Journal of the American Statistical Association 57(298), 387–402 (1962)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Robert Görke
    • 1
  • Roland Kluge
    • 1
  • Andrea Schumm
    • 1
  • Christian Staudt
    • 1
  • Dorothea Wagner
    • 1
  1. 1.Institute of Theoretical InformaticsKarlsruhe Institute of TechnologyGermany

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