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A Quality Metric for Visualization of Clusters in Graphs

Part of the Lecture Notes in Computer Science book series (LNTCS,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.

This work is supported by ARC DP grant.

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References

  1. Aldenderfer, M.S., Blashfield, R.: Cluster Analysis. Beverly Hills: Sage Publications, Thousand Oaks (1984)

    CrossRef  Google Scholar 

  2. Batagelj, V., Mrvar, A.: Pajek data sets (2003). http://pajek.imfm.si/doku.php?id=data:index

  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. 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_47

    CrossRef  Google Scholar 

  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. 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_6

    CrossRef  MATH  Google Scholar 

  7. Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley-Interscience, New York (1991)

    Google Scholar 

  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_34

    CrossRef  Google Scholar 

  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)

    MathSciNet  CrossRef  Google Scholar 

  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_57

    CrossRef  MATH  Google Scholar 

  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

    CrossRef  Google Scholar 

  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/10.1080/01621459.1983.10478008

    CrossRef  MATH  Google Scholar 

  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

    Google Scholar 

  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_25

    CrossRef  MATH  Google Scholar 

  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_29

    CrossRef  MATH  Google Scholar 

  16. Hu, Y.: Efficient, high-quality force-directed graph drawing. Math. J. 10(1), 37–71 (2005)

    Google Scholar 

  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. Hubert, L., Arabie, P.: Comparing partitions. J. Classif. 2(1), 193–218 (1985). https://doi.org/10.1007/BF01908075

    CrossRef  MATH  Google Scholar 

  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_20

    CrossRef  MATH  Google Scholar 

  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.2004.08.015

    MathSciNet  CrossRef  MATH  Google Scholar 

  21. Kruiger, J.F.: tsnet (2017). https://github.com/HanKruiger/tsNET/

  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.13187

    CrossRef  Google Scholar 

  23. Leskovec, J., Krevl, A.: SNAP Datasets: Stanford large network dataset collection, June 2014. http://snap.stanford.edu/data

  24. Maaten, L.V.D., Hinton, G.: Visualizing data using t-SNE. J. Mach. Learn. Res. 9(Nov), 2579–2605 (2008)

    Google Scholar 

  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. 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_40

    CrossRef  MATH  Google Scholar 

  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.00370

    MathSciNet  CrossRef  MATH  Google Scholar 

  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_2

    CrossRef  Google Scholar 

  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. 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_67

    CrossRef  Google Scholar 

  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/BFb0021827

    CrossRef  Google Scholar 

  32. Rand, W.M.: Objective criteria for the evaluation of clustering methods. J. Am. Stat. Assoc. 66(336), 846–850 (1971). https://doi.org/10.1080/01621459.1971.10482356

    CrossRef  Google Scholar 

  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. Saket, B., Simonetto, P., Kobourov, S.: Group-level graph visualization taxonomy. CoRR abs/1403.7421 (2014)

    Google Scholar 

  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

    CrossRef  Google Scholar 

  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. Torgerson, W.S.: Multidimensional scaling: I. Theory and method. Psychometrika 17(4), 401–419 (1952). https://doi.org/10.1007/BF02288916

    MathSciNet  CrossRef  Google Scholar 

  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. 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

    CrossRef  Google Scholar 

  40. Zitnik, M., Sosič, R., Maheshwari, S., Leskovec, J.: BioSNAP Datasets: Stanford biomedical network dataset collection, August 2018. http://snap.stanford.edu/biodata

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Correspondence to Amyra Meidiana .

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Meidiana, A., Hong, SH., Eades, P., Keim, D. (2019). A Quality Metric for Visualization of Clusters in Graphs. In: Archambault, D., Tóth, C. (eds) Graph Drawing and Network Visualization. GD 2019. Lecture Notes in Computer Science(), vol 11904. Springer, Cham. https://doi.org/10.1007/978-3-030-35802-0_10

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