Data Mining and Knowledge Discovery

, Volume 27, Issue 1, pp 84–116

A regularized graph layout framework for dynamic network visualization

Article

Abstract

Many real-world networks, including social and information networks, are dynamic structures that evolve over time. Such dynamic networks are typically visualized using a sequence of static graph layouts. In addition to providing a visual representation of the network structure at each time step, the sequence should preserve the mental map between layouts of consecutive time steps to allow a human to interpret the temporal evolution of the network. In this paper, we propose a framework for dynamic network visualization in the on-line setting where only present and past graph snapshots are available to create the present layout. The proposed framework creates regularized graph layouts by augmenting the cost function of a static graph layout algorithm with a grouping penalty, which discourages nodes from deviating too far from other nodes belonging to the same group, and a temporal penalty, which discourages large node movements between consecutive time steps. The penalties increase the stability of the layout sequence, thus preserving the mental map. We introduce two dynamic layout algorithms within the proposed framework, namely dynamic multidimensional scaling and dynamic graph Laplacian layout. We apply these algorithms on several data sets to illustrate the importance of both grouping and temporal regularization for producing interpretable visualizations of dynamic networks.

Keywords

Graph layout Dynamic network Visualization Mental map Regularization Multidimensional scaling Spectral layout Graph Laplacian 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baur M, Schank T (2008) Dynamic graph drawing in Visone. Tech. rep., Universität KarlsruheGoogle Scholar
  2. Bazaraa MS, Sherali HD, Shetty CM (2006) Nonlinear programming: theory and algorithms. Wiley, New YorkMATHCrossRefGoogle Scholar
  3. Belkin M, Niyogi P (2003) Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput 15(6): 1373–1396MATHCrossRefGoogle Scholar
  4. Bender-deMoll S, McFarland DA (2006) The art and science of dynamic network visualization. J Soc Struct 7(2): 1–38Google Scholar
  5. Bender-deMoll S, McFarland DA (2012) SoNIA—Social Network Image Animator. http://www.stanford.edu/group/sonia/
  6. Borg I, Groenen PJF (2005) Modern multidimensional scaling. Springer, New YorkGoogle Scholar
  7. Brandes U, Corman SR (2003) Visual unrolling of network evolution and the analysis of dynamic discourse. Inf Vis 2(1): 40–50CrossRefGoogle Scholar
  8. Brandes U, Mader M (2011) A quantitative comparison of stress-minimization approaches for offline dynamic graph drawing. In: Proceedings of the 19th international symposium on graph drawing, pp 99–110Google Scholar
  9. Brandes U, Wagner D (1997) A Bayesian paradigm for dynamic graph layout. In: Proceedings of the 5th international symposium on graph drawing, pp 236–247Google Scholar
  10. Brandes U, Wagner D (2004) visone—analysis and visualization of social networks. In: Jünger M, Mutzel P (eds) Graph drawing software. Springer, Berlin, pp 321–340CrossRefGoogle Scholar
  11. Brandes U, Fleischer D, Puppe T (2007) Dynamic spectral layout with an application to small worlds. J Graph Algorithms Appl 11(2): 325–343MathSciNetMATHCrossRefGoogle Scholar
  12. Brandes U, Indlekofer N, Mader M (2012) Visualization methods for longitudinal social networks and stochastic actor-oriented modeling. Soc Netw 34(3): 291–308CrossRefGoogle Scholar
  13. Branke J (2001) Dynamic graph drawing. In: Kaufmann M, Wagner D (eds) Drawing graphs: methods and models. Springer, Berlin, pp 228–246CrossRefGoogle Scholar
  14. Byrd RH, Hribar ME, Nocedal J (1999) An interior point algorithm for large-scale nonlinear programming. SIAM J Optim 9(4): 877–900MathSciNetMATHCrossRefGoogle Scholar
  15. Chi Y, Song X, Zhou D, Hino K, Tseng BL (2009) On evolutionary spectral clustering. ACM Trans Knowl Discov Data 3(4): 17CrossRefGoogle Scholar
  16. Costa JA, Hero III AO (2005) Classification constrained dimensionality reduction. In: Proceedings of the IEEE international conference on acoustics, speech, and signal processing, pp 1077–1080Google Scholar
  17. de Leeuw J, Heiser WJ (1980) Multidimensional scaling with restrictions on the configuration. In: Proceedings of the 5th international symposium on multivariate analysis, pp 501–522Google Scholar
  18. Di Battista G, Eades P, Tamassia R, Tollis IG (1999) Graph drawing: algorithms for the visualization of graphs. Prentice Hall, Upper Saddle RiverMATHGoogle Scholar
  19. Eades P, Huang ML (2000) Navigating clustered graphs using force-directed methods. J Graph Algorithms Appl 4(3): 157–181MATHCrossRefGoogle Scholar
  20. Eagle N, Pentland A, Lazer D (2009) Inferring friendship network structure by using mobile phone data. Proc Natl Acad Sci USA 106(36): 15274–15278CrossRefGoogle Scholar
  21. Erten C, Harding PJ, Kobourov SG, Wampler K, Yee G (2004) Exploring the computing literature using temporal graph visualization. In: Proceedings of the conference on visualization and data analysis, pp 45–56Google Scholar
  22. Frishman Y, Tal A (2008) Online dynamic graph drawing. IEEE Trans Vis Comput Graphics 14(4): 727–740CrossRefGoogle Scholar
  23. Fruchterman TMJ, Reingold EM (1991) Graph drawing by force-directed placement. Softw Pract Exp 21(11): 1129–1164CrossRefGoogle Scholar
  24. Gansner ER, Koren Y, North S (2004) Graph drawing by stress majorization. In: Proceedings of the 12th international symposium on graph drawings, pp 239–250Google Scholar
  25. Hall KM (1970) An r-dimensional quadratic placement algorithm. Manag Sci 17(3): 219–229MATHCrossRefGoogle Scholar
  26. Herman I, Melançon G, Marshall MS (2000) Graph visualisation and navigation in information visualisation: a survey. IEEE Trans Vis Comput Graphics 6(1): 24–43CrossRefGoogle Scholar
  27. Hoerl AE, Kennard RW (1970) Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12(1): 55–67MathSciNetMATHCrossRefGoogle Scholar
  28. Holland PW, Laskey KB, Leinhardt S (1983) Stochastic blockmodels: first steps. Soc Netw 5(2): 109–137MathSciNetCrossRefGoogle Scholar
  29. Kamada T, Kawai S (1989) An algorithm for drawing general undirected graphs. Inf Process Lett 31(12): 7–15MathSciNetMATHCrossRefGoogle Scholar
  30. Koren Y (2005) Drawing graphs by eigenvectors: theory and practice. Comput Math Appl 49(11–12): 1867–1888MathSciNetMATHCrossRefGoogle Scholar
  31. Kossinets G, Watts DJ (2006) Empirical analysis of an evolving social network. Science 311(5757): 88–90MathSciNetMATHCrossRefGoogle Scholar
  32. Lee JA, Verleysen M (2007) Nonlinear dimensionality reduction. Springer, BerlinMATHCrossRefGoogle Scholar
  33. Leskovec J, Kleinberg J, Faloutsos C (2007) Graph evolution: densification and shrinking diameters. ACM Trans Knowl Discov Data 1(1): 2CrossRefGoogle Scholar
  34. Leydesdorff L, Schank T (2008) Dynamic animations of journal maps: indicators of structural changes and interdisciplinary developments. J Am Soc Inf Sci Technol 59(11): 1810–1818CrossRefGoogle Scholar
  35. Lütkepohl H (1997) Handbook of matrices. Wiley, New YorkGoogle Scholar
  36. Misue K, Eades P, Lai W, Sugiyama K (1995) Layout adjustment and the mental map. J Vis Lang Comput 6(2): 183–210CrossRefGoogle Scholar
  37. MIT-WWW (2005) MIT Academic Calendar 2004–2005. http://web.mit.edu/registrar/www/calendar0405.html
  38. Moody J, McFarland D, Bender-deMoll S (2005) Dynamic network visualization. Am J Sociol 110(4): 1206–1241CrossRefGoogle Scholar
  39. Mucha PJ, Richardson T, Macon K, Porter MA, Onnela JP (2010) Community structure in time-dependent, multiscale, and multiplex networks. Science 328(5980): 876–878MathSciNetMATHCrossRefGoogle Scholar
  40. Newcomb TM (1961) The acquaintance process. Holt, Rinehart and Winston, New YorkCrossRefGoogle Scholar
  41. Ng AY, Jordan MI, Weiss Y (2001) On spectral clustering: analysis and an algorithm. Adv. Neural Inf. Process. Syst. 14: 849–856Google Scholar
  42. Nordlie PG (1958) A longitudinal study of interpersonal attraction in a natural group setting. PhD thesis, University of MichiganGoogle Scholar
  43. Sun J, Xie Y, Zhang H, Faloutsos C (2007) Less is more: compact matrix decomposition for large sparse graphs. In: Proceedings of the 7th SIAM conference on data mining, pp 366–377Google Scholar
  44. Tibshirani R (1996) Regression shrinkage and selection via the lasso. J R Stat Soc Ser B 58(1): 267–288MathSciNetMATHGoogle Scholar
  45. Tong H, Papadimitriou S, Sun J, Yu PS, Faloutsos C (2008) Colibri: fast mining of large static and dynamic graphs. In: Proceedings of the 14th ACM SIGKDD international conference on knowledge discovery and data mining, pp 686–694Google Scholar
  46. Trefethen LN, Bau D III (1997) Numerical linear algebra. SIAM, PhiladelphiaMATHCrossRefGoogle Scholar
  47. Visone-WWW (2012) visone. http://www.visone.info/
  48. Wang X, Miyamoto I (1995) Generating customized layouts. In: Proceedings of the symposium on graph drawing, pp 504–515Google Scholar
  49. Witten DM, Tibshirani R (2011) Supervised multidimensional scaling for visualization, classification, and bipartite ranking. Comput Stat Data Anal 55(1): 789–801MathSciNetMATHCrossRefGoogle Scholar
  50. Witten DM, Tibshirani R, Hastie T (2009) A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Biostatistics 10(3): 515–534CrossRefGoogle Scholar
  51. Xu KS, Kliger M, Hero III AO (2011a) Adaptive evolutionary clustering (submitted). arXiv:1104.1990Google Scholar
  52. Xu KS, Kliger M, Hero III AO (2011b) Visualizing the temporal evolution of dynamic networks. In: Proceedings of the 9th workshop on mining and learning graphsGoogle Scholar
  53. Xu KS, Kliger M, Hero III AO (2012) A regularized graph layout framework for dynamic network visualization: supporting website. http://tbayes.eecs.umich.edu/xukevin/visualization_dmkd_2012

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.EECS DepartmentUniversity of MichiganAnn ArborUSA
  2. 2.Omek Interactive, Ltd.Beit ShemeshIsrael

Personalised recommendations