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.
Similar content being viewed by others
References
Baur M, Schank T (2008) Dynamic graph drawing in Visone. Tech. rep., Universität Karlsruhe
Bazaraa MS, Sherali HD, Shetty CM (2006) Nonlinear programming: theory and algorithms. Wiley, New York
Belkin M, Niyogi P (2003) Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput 15(6): 1373–1396
Bender-deMoll S, McFarland DA (2006) The art and science of dynamic network visualization. J Soc Struct 7(2): 1–38
Bender-deMoll S, McFarland DA (2012) SoNIA—Social Network Image Animator. http://www.stanford.edu/group/sonia/
Borg I, Groenen PJF (2005) Modern multidimensional scaling. Springer, New York
Brandes U, Corman SR (2003) Visual unrolling of network evolution and the analysis of dynamic discourse. Inf Vis 2(1): 40–50
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 99–110
Brandes U, Wagner D (1997) A Bayesian paradigm for dynamic graph layout. In: Proceedings of the 5th international symposium on graph drawing, pp 236–247
Brandes U, Wagner D (2004) visone—analysis and visualization of social networks. In: Jünger M, Mutzel P (eds) Graph drawing software. Springer, Berlin, pp 321–340
Brandes U, Fleischer D, Puppe T (2007) Dynamic spectral layout with an application to small worlds. J Graph Algorithms Appl 11(2): 325–343
Brandes U, Indlekofer N, Mader M (2012) Visualization methods for longitudinal social networks and stochastic actor-oriented modeling. Soc Netw 34(3): 291–308
Branke J (2001) Dynamic graph drawing. In: Kaufmann M, Wagner D (eds) Drawing graphs: methods and models. Springer, Berlin, pp 228–246
Byrd RH, Hribar ME, Nocedal J (1999) An interior point algorithm for large-scale nonlinear programming. SIAM J Optim 9(4): 877–900
Chi Y, Song X, Zhou D, Hino K, Tseng BL (2009) On evolutionary spectral clustering. ACM Trans Knowl Discov Data 3(4): 17
Costa JA, Hero III AO (2005) Classification constrained dimensionality reduction. In: Proceedings of the IEEE international conference on acoustics, speech, and signal processing, pp 1077–1080
de Leeuw J, Heiser WJ (1980) Multidimensional scaling with restrictions on the configuration. In: Proceedings of the 5th international symposium on multivariate analysis, pp 501–522
Di Battista G, Eades P, Tamassia R, Tollis IG (1999) Graph drawing: algorithms for the visualization of graphs. Prentice Hall, Upper Saddle River
Eades P, Huang ML (2000) Navigating clustered graphs using force-directed methods. J Graph Algorithms Appl 4(3): 157–181
Eagle N, Pentland A, Lazer D (2009) Inferring friendship network structure by using mobile phone data. Proc Natl Acad Sci USA 106(36): 15274–15278
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 45–56
Frishman Y, Tal A (2008) Online dynamic graph drawing. IEEE Trans Vis Comput Graphics 14(4): 727–740
Fruchterman TMJ, Reingold EM (1991) Graph drawing by force-directed placement. Softw Pract Exp 21(11): 1129–1164
Gansner ER, Koren Y, North S (2004) Graph drawing by stress majorization. In: Proceedings of the 12th international symposium on graph drawings, pp 239–250
Hall KM (1970) An r-dimensional quadratic placement algorithm. Manag Sci 17(3): 219–229
Herman I, Melançon G, Marshall MS (2000) Graph visualisation and navigation in information visualisation: a survey. IEEE Trans Vis Comput Graphics 6(1): 24–43
Hoerl AE, Kennard RW (1970) Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12(1): 55–67
Holland PW, Laskey KB, Leinhardt S (1983) Stochastic blockmodels: first steps. Soc Netw 5(2): 109–137
Kamada T, Kawai S (1989) An algorithm for drawing general undirected graphs. Inf Process Lett 31(12): 7–15
Koren Y (2005) Drawing graphs by eigenvectors: theory and practice. Comput Math Appl 49(11–12): 1867–1888
Kossinets G, Watts DJ (2006) Empirical analysis of an evolving social network. Science 311(5757): 88–90
Lee JA, Verleysen M (2007) Nonlinear dimensionality reduction. Springer, Berlin
Leskovec J, Kleinberg J, Faloutsos C (2007) Graph evolution: densification and shrinking diameters. ACM Trans Knowl Discov Data 1(1): 2
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): 1810–1818
Lütkepohl H (1997) Handbook of matrices. Wiley, New York
Misue K, Eades P, Lai W, Sugiyama K (1995) Layout adjustment and the mental map. J Vis Lang Comput 6(2): 183–210
MIT-WWW (2005) MIT Academic Calendar 2004–2005. http://web.mit.edu/registrar/www/calendar0405.html
Moody J, McFarland D, Bender-deMoll S (2005) Dynamic network visualization. Am J Sociol 110(4): 1206–1241
Mucha PJ, Richardson T, Macon K, Porter MA, Onnela JP (2010) Community structure in time-dependent, multiscale, and multiplex networks. Science 328(5980): 876–878
Newcomb TM (1961) The acquaintance process. Holt, Rinehart and Winston, New York
Ng AY, Jordan MI, Weiss Y (2001) On spectral clustering: analysis and an algorithm. Adv. Neural Inf. Process. Syst. 14: 849–856
Nordlie PG (1958) A longitudinal study of interpersonal attraction in a natural group setting. PhD thesis, University of Michigan
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 366–377
Tibshirani R (1996) Regression shrinkage and selection via the lasso. J R Stat Soc Ser B 58(1): 267–288
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 686–694
Trefethen LN, Bau D III (1997) Numerical linear algebra. SIAM, Philadelphia
Visone-WWW (2012) visone. http://www.visone.info/
Wang X, Miyamoto I (1995) Generating customized layouts. In: Proceedings of the symposium on graph drawing, pp 504–515
Witten DM, Tibshirani R (2011) Supervised multidimensional scaling for visualization, classification, and bipartite ranking. Comput Stat Data Anal 55(1): 789–801
Witten DM, Tibshirani R, Hastie T (2009) A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Biostatistics 10(3): 515–534
Xu KS, Kliger M, Hero III AO (2011a) Adaptive evolutionary clustering (submitted). arXiv:1104.1990
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 graphs
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
Author information
Authors and Affiliations
Corresponding author
Additional information
Responsible editor: Barbara Hammer; Daniel Keim; Guy Lebanon; Neil Lawrence.
Rights and permissions
About this article
Cite this article
Xu, K.S., Kliger, M. & Hero, A.O. A regularized graph layout framework for dynamic network visualization. Data Min Knowl Disc 27, 84–116 (2013). https://doi.org/10.1007/s10618-012-0286-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10618-012-0286-6