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A Quality Metric for Visualization of Clusters in Graphs

  • Amyra MeidianaEmail author
  • Seok-Hee Hong
  • Peter Eades
  • Daniel Keim
Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11904)

Abstract

Traditionally, graph quality metrics focus on readability, but recent studies show the need for metrics which are more specific to the discovery of patterns in graphs. Cluster analysis is a popular task within graph analysis, yet there is no metric yet explicitly quantifying how well a drawing of a graph represents its cluster structure.

We define a clustering quality metric measuring how well a node-link drawing of a graph represents the clusters contained in the graph. Experiments with deforming graph drawings verify that our metric effectively captures variations in the visual cluster quality of graph drawings. We then use our metric to examine how well different graph drawing algorithms visualize cluster structures in various graphs; the results confirm that some algorithms which have been specifically designed to show cluster structures perform better than other algorithms.

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References

  1. 1.
    Aldenderfer, M.S., Blashfield, R.: Cluster Analysis. Beverly Hills: Sage Publications, Thousand Oaks (1984)Google Scholar
  2. 2.
    Batagelj, V., Mrvar, A.: Pajek data sets (2003). http://pajek.imfm.si/doku.php?id=data:index
  3. 3.
    Battista, G.D., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall PTR, Upper Saddle River (1998)Google Scholar
  4. 4.
    Baur, M., Benkert, M., Brandes, U., Cornelsen, S., Gaertler, M., Köpf, B., Lerner, J., Wagner, D.: Visone Software for Visual Social Network Analysis. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 463–464. Springer, Heidelberg (2002).  https://doi.org/10.1007/3-540-45848-4_47CrossRefGoogle Scholar
  5. 5.
    Behrisch, M., Blumenschein, M., Kim, N.W., Shao, L., El-Assady, M., Fuchs, J., Seebacher, D., Diehl, A., Brandes, U., Pfister, H., Schreck, T., Weiskopf, D., Keim, D.A.: Quality metrics for information visualization. In: Computer Graphics Forum, vol. 37, pp. 625–662. Wiley Online Library (2018)Google Scholar
  6. 6.
    Brandes, U., Pich, C.: Eigensolver methods for progressive multidimensional scaling of large data. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 42–53. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-70904-6_6CrossRefzbMATHGoogle Scholar
  7. 7.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley-Interscience, New York (1991)Google Scholar
  8. 8.
    David, A.: Tulip. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 435–437. Springer, Heidelberg (2002).  https://doi.org/10.1007/3-540-45848-4_34CrossRefGoogle Scholar
  9. 9.
    Eades, P., Hong, S.H., Nguyen, A., Klein, K.: Shape-based quality metrics for large graph visualization. J. Graph Algorithms Appl. 21(1), 29–53 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ellson, J., Gansner, E., Koutsofios, L., North, S.C., Woodhull, G.: Graphviz— open source graph drawing tools. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 483–484. Springer, Heidelberg (2002).  https://doi.org/10.1007/3-540-45848-4_57CrossRefzbMATHGoogle Scholar
  11. 11.
    Estivill-Castro, V.: Why so many clustering algorithms: a position paper. SIGKDD Explor. Newsl. 4(1), 65–75 (2002).  https://doi.org/10.1145/568574.568575
  12. 12.
    Fowlkes, E.B., Mallows, C.L.: A method for comparing two hierarchical clusterings. J. Am. Stat. Assoc. 78(383), 553–569 (1983).  https://doi.org/10.1080/01621459.1983.10478008CrossRefzbMATHGoogle Scholar
  13. 13.
    Fruchterman, T.M.J., Reingold, E.M.: Graph drawing by force-directed placement. Softw.: Practice Exp. 21(11), 1129–1164 (1991).  https://doi.org/10.1002/spe.4380211102
  14. 14.
    Gansner, E.R., Koren, Y., North, S.: Graph drawing by stress majorization. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 239–250. Springer, Heidelberg (2005).  https://doi.org/10.1007/978-3-540-31843-9_25CrossRefzbMATHGoogle Scholar
  15. 15.
    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).  https://doi.org/10.1007/978-3-540-31843-9_29CrossRefzbMATHGoogle Scholar
  16. 16.
    Hu, Y.: Efficient, high-quality force-directed graph drawing. Math. J. 10(1), 37–71 (2005)Google Scholar
  17. 17.
    Huang, W., Hong, S.H., Eades, P.: Effects of crossing angles. In: 2008 IEEE Pacific Visualization Symposium, pp. 41–46. IEEE (2008)Google Scholar
  18. 18.
    Hubert, L., Arabie, P.: Comparing partitions. J. Classif. 2(1), 193–218 (1985).  https://doi.org/10.1007/BF01908075CrossRefzbMATHGoogle Scholar
  19. 19.
    Kobourov, S.G., Pupyrev, S., Saket, B.: Are crossings important for drawing large graphs? In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 234–245. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-45803-7_20CrossRefzbMATHGoogle Scholar
  20. 20.
    Koren, Y.: Drawing graphs by eigenvectors: theory and practice. Comput. Math. Appl. 49(11–12), 1867–1888 (2005).  https://doi.org/10.1016/j.camwa.2004.08.015MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kruiger, J.F.: tsnet (2017). https://github.com/HanKruiger/tsNET/
  22. 22.
    Kruiger, J.F., Rauber, P.E., Martins, R.M., Kerren, A., Kobourov, S., Telea, A.C.: Graph layouts by t-SNE. Comput. Graph. Forum 36(3), 283–294 (2017).  https://doi.org/10.1111/cgf.13187CrossRefGoogle Scholar
  23. 23.
    Leskovec, J., Krevl, A.: SNAP Datasets: Stanford large network dataset collection, June 2014. http://snap.stanford.edu/data
  24. 24.
    Maaten, L.V.D., Hinton, G.: Visualizing data using t-SNE. J. Mach. Learn. Res. 9(Nov), 2579–2605 (2008)Google Scholar
  25. 25.
    MacQueen, J., et al.: Some methods for classification and analysis of multivariate observations. In: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 281–297. University of California Press (1967)Google Scholar
  26. 26.
    Noack, A.: An energy model for visual graph clustering. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 425–436. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-24595-7_40CrossRefzbMATHGoogle Scholar
  27. 27.
    Nocaj, A., Ortmann, M., Brandes, U.: Untangling the hairballs of multi-centered, small-world online social media networks. J. Graph Algorithms Appl. 19(2), 595–618 (2015).  https://doi.org/10.7155/jgaa.00370MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ortmann, M., Klimenta, M., Brandes, U.: A sparse stress model. In: Hu, Y., Nöllenburg, M. (eds.) GD 2016. LNCS, vol. 9801, pp. 18–32. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-50106-2_2CrossRefGoogle Scholar
  29. 29.
    Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: machine learning in python. J. Mach. Learn. Res. 12(Oct), 2825–2830 (2011)Google Scholar
  30. 30.
    Purchase, H.: Which aesthetic has the greatest effect on human understanding? In: DiBattista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 248–261. Springer, Heidelberg (1997).  https://doi.org/10.1007/3-540-63938-1_67CrossRefGoogle Scholar
  31. 31.
    Purchase, H.C., Cohen, R.F., James, M.: Validating graph drawing aesthetics. In: Brandenburg, F.J. (ed.) GD 1995. LNCS, vol. 1027, pp. 435–446. Springer, Heidelberg (1996).  https://doi.org/10.1007/BFb0021827CrossRefGoogle Scholar
  32. 32.
    Rand, W.M.: Objective criteria for the evaluation of clustering methods. J. Am. Stat. Assoc. 66(336), 846–850 (1971).  https://doi.org/10.1080/01621459.1971.10482356CrossRefGoogle Scholar
  33. 33.
    Rosenberg, A., Hirschberg, J.: V-measure: a conditional entropy-based external cluster evaluation measure. In: Proceedings of the 2007 Joint Conference on Empirical Methods in Natural Language Processing and Computational Natural Language Learning (EMNLP-CoNLL), pp. 410–420 (2007)Google Scholar
  34. 34.
    Saket, B., Simonetto, P., Kobourov, S.: Group-level graph visualization taxonomy. CoRR abs/1403.7421 (2014)Google Scholar
  35. 35.
    Sedlmair, M., Tatu, A., Munzner, T., Tory, M.: A taxonomy of visual cluster separation factors. Comput. Graph. Forum 31(3pt4), 1335–1344 (2012).  https://doi.org/10.1111/j.1467-8659.2012.03125.x
  36. 36.
    Strehl, A., Ghosh, J.: Cluster ensembles–a knowledge reuse framework for combining multiple partitions. J. Mach. Learn. Res. 3(Dec), 583–617 (2002)Google Scholar
  37. 37.
    Torgerson, W.S.: Multidimensional scaling: I. Theory and method. Psychometrika 17(4), 401–419 (1952).  https://doi.org/10.1007/BF02288916
  38. 38.
    Vinh, N.X., Epps, J., Bailey, J.: Information theoretic measures for clusterings comparison: variants, properties, normalization and correction for chance. J. Mach. Learn. Res. 11(Oct), 2837–2854 (2010)Google Scholar
  39. 39.
    Wiese, R., Eiglsperger, M., Kaufmann, M.: yfiles - visualization and automatic layout of graphs. In: Jünger, M., Mutzel, P. (eds.) Graph Drawing Software. Mathematics and Visualization, pp. 173–191. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-642-18638-7_8
  40. 40.
    Zitnik, M., Sosič, R., Maheshwari, S., Leskovec, J.: BioSNAP Datasets: Stanford biomedical network dataset collection, August 2018. http://snap.stanford.edu/biodata

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Amyra Meidiana
    • 1
    Email author
  • Seok-Hee Hong
    • 1
  • Peter Eades
    • 1
  • Daniel Keim
    • 2
  1. 1.University of SydneySydneyAustralia
  2. 2.University of KonstanzKonstanzGermany

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