Skip to main content
SpringerLink
Log in

Inverse Markov Process Based Constrained Dynamic Graph Layout

  • Regular Paper
  • Published:
Journal of Computer Science and Technology Aims and scope Submit manuscript

Abstract

In online dynamic graph drawing, constraints over nodes and node pairs help preserve a coherent mental map in a sequence of graphs. Defining the constraints is challenging due to the requirements of both preserving mental map and satisfying the visual aesthetics of a graph layout. Most existing algorithms basically depend on local changes but fail to do proper evaluations on the global propagation when setting constraints. To solve this problem, we introduce a heuristic model derived from PageRank which simulates the node movement as an inverse Markov process hence to give a global analysis of the layout's change, according to which different constraints can be set. These constraints, along with stress function, generate layouts maintaining spatial positions and shapes of relatively stable substructures between adjacent graphs. Experiments demonstrate that our method preserves both structure and position similarity to help users track graph changes visually.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Frishman Y, Tal A. Online dynamic graph drawing. IEEE Transactions on Visualization and Computer Graphics, 2008, 14(4): 727-740. DOI: https://doi.org/10.1109/TVCG.2008.11.

    Article  Google Scholar 

  2. Branke J. Dynamic graph drawing. In Drawing Graphs, Kaufmann M, Wagner D (eds.), Springer, 2001, pp.228-246. DOI: 10.1007/3-540-44969-8_9.

  3. Che L, Liang J, Yuan X et al. Laplacian-based dynamic graph visualization. In Proc. the 2015 IEEE Pacific Visualization Symposium, April 2015, pp.69-73. DOI: 10.1109/PACIFICVIS.2015.7156358.

  4. Gorochowski T E, Bernardo M D, Grierson C S. Using aging to visually uncover evolutionary processes on networks. IEEE Transactions on Visualization and Computer Graphics, 2011, 18(8): 1343-1352. DOI: https://doi.org/10.1109/TVCG.2011.142.

    Article  Google Scholar 

  5. Feng K C, Wang C, Shen H W et al. Coherent time-varying graph drawing with multifocus + context interaction. IEEE Transactions on Visualization and Computer Graphics, 2012, 18(8): 1330-1342. DOI: https://doi.org/10.1109/TVCG.2011.128.

    Article  Google Scholar 

  6. Wang Y, Wang Y, Sun Y et al. Revisiting stress majorization as a unified framework for interactive constrained graph visualization. IEEE Transactions on Visualization and Computer Graphics, 2018, 24(1): 489-499. DOI: https://doi.org/10.1109/TVCG.2017.2745919.

    Article  Google Scholar 

  7. Yuan X R, Chen L M, Hu Y F et al. Intelligent graph layout using many users' input. IEEE Transactions on Visualization and Computer Graphics, 2012, 18(12): 2699-2708. DOI: https://doi.org/10.1109/TVCG.2012.236.

    Article  Google Scholar 

  8. Kamada T, Kawai S. An algorithm for drawing general undirected graphs. Information Processing Letters, 1989, 31(1): 7-15. DOI: https://doi.org/10.1016/0020-0190(89)90102-6.

    Article  MathSciNet  MATH  Google Scholar 

  9. Crnovrsanin T, Chu J, Ma K L. An incremental layout method for visualizing online dynamic graphs. In Proc. the 23rd International Symposium on Graph Drawing and Network Visualization, Sept. 2015, pp.16-29. DOI: 10.1007/978-3-319-27261-0_2.

  10. Kaufmann M, Dorothea W. Drawing Graphs: Methods and Models. Springer, 2001. DOI: https://doi.org/10.1007/3-540-44969-8.

  11. Eades P. A heuristic for graph drawing. Congressus Nutnerantiunt, 1984, 42: 149-160.

    MathSciNet  Google Scholar 

  12. Fruchterman T M, Reingold E M. Graph drawing by forcedirected placement. Software: Practice and Experience, 1991, 21(11): 1129-64. DOI: https://doi.org/10.1002/spe.4380211102.

    Article  Google Scholar 

  13. Cohen R F, Battista G D, Tamassia R, Tollis I G, Bertolazzi P. A framework for dynamic graph drawing. In Proc. the 8th Annual Symposium on Computational Geometry, July 1992, pp.261-270. DOI: 10.1145/142675.142728.

  14. Misue K, Eades P, Lai W et al. Layout adjustment and the mental map. Journal of Visual Languages & Computing, 1995, 6(2): 183-210. DOI: https://doi.org/10.1006/jvlc.1995.1010.

    Article  Google Scholar 

  15. Eades P, Cohen R F, Huang M L. Online animated graph drawing for web navigation. In Proc. the 5th International Symposium on Graph Drawing, Sept. 1997, pp.330-335. DOI: 10.1007/3-540-63938-1_77.

  16. Erten C, Harding P J, Kobourov S G, Wampler K, Yee G. GraphAEL: Graph animations with evolving layouts. In Proc. the 11th International Symposium on Graph Drawing, Sept. 2003, pp.98-110. DOI: 10.1007/978-3-540-24595-7_9.

  17. Bridgernan S, Tamassia R. Difference metrics for interactive orthogonal graph drawing algorithms. In Proc. the 6th International Symposium on Graph Drawing, August 1998, pp.57-71. DOI: 10.1007/3-540-37623-2_5.

  18. Beck F, Burch M, Diehl S, Weiskopf D. The state of the art in visualizing dynamic graphs. In Proc. the 16th Eurographics Conference on Visualization, June 2014. DOI: 10.2312/eurovisstar.20141174.

  19. Archambault D, Purchase H, Pinaud B. Animation, small multiples, and the effect of mental map preservation in dynamic graphs. IEEE Transactions on Visualization and Computer Graphics, 2010, 17(4): 539-552. DOI: https://doi.org/10.1109/TVCG.2010.78.

    Article  Google Scholar 

  20. Jacomy M, Venturini T, Heymann S et al. ForceAtlas2, a continuous graph layout algorithm for handy network visualization designed for the Gephi software. PLoS ONE, 2014, 9(6): Article No. e98679. DOI: 10.1371/journal.pone.0098679.

  21. Friedrich C, Eades P. The Marey graph animation tool demo. In Proc. the 8th International Symposium on Graph Drawing, Sept. 2000, pp.396-406. DOI: 10.1007/3-540-44541-2_37.

  22. Friedrich C, Eades P. Graph drawing in motion. J. Graph Algorithms Appl., 2002, 6(3): 353-70. DOI: https://doi.org/10.1007/3-540-45848-4_18.

    Article  MathSciNet  MATH  Google Scholar 

  23. Hurter C, Ersoy O, Fabrikant S I, Klein T R, Telea A C. Bundled visualization of dynamic graph and trail data. IEEE Transactions on Visualization and Computer Graphics, 2013, 20(8): 1141-57. DOI: https://doi.org/10.1109/TVCG.2013.246.

    Article  Google Scholar 

  24. Diehl S, Görg C, Kerren A. Preserving the mental map using foresighted layout. In Proc. the Joint Eurographics | IEEE TCVG Symposium on Visualization, May 2001, pp.175-184. DOI: 10.1007/978-3-7091-6215-6_19.

  25. Diehl S, Görg C. Graphs, they are changing. In Proc. the 10th International Symposium on Graph Drawing, August 2002, pp.23-30. DOI: 10.1007/3-540-36151-0_3.

  26. Brandes U, Wagner D. A Bayesian paradigm for dynamic graph layout. In Proc. the 5th International Symposium on Graph Drawing, Sept. 1997, pp.236-247. DOI: 10.1007/3-540-63938-1_66.

  27. Görg C, Birke P, Pohl M, Diehl S. Dynamic graph drawing of sequences of orthogonal and hierarchical graphs. In Proc. the 12th International Symposium on Graph Drawing, Sept. 2004, pp.228-238. DOI: 10.1007/978-3-540-31843-9_24.

  28. Lee Y Y, Lin C C, Yen H C. Mental map preserving graph drawing using simulated annealing. In Proc. the 2006 Asia-Pacific Symposium on Information Visualisation, Jan. 2006, pp.179-188. DOI: 10.1016/j.ins.2011.06.005.

  29. Crnovrsanin T, Chu J, Ma K L. An incremental layout method for visualizing online dynamic graphs. In Proc. the 23rd International Symposium on Graph Drawing, Sept. 2015, pp.16-29. DOI: 10.1007/978-3-319-27261-0_2.

  30. Hachul S, Jünger M. Drawing large graphs with a potential-field-based multilevel algorithm. In Proc. the 12th International Symposium on Graph Drawing, Sept. 2004, pp.285-295. DOI: 10.1007/978-3-540-31843-9_29.

  31. Dwyer T, Koren Y. DIG-COLA: Directed graph layout through constrained energy minimization. In Proc. the 2005 IEEE Symposium on Information Visualization, October 2005, pp.65-72. DOI: 10.1109/INFVIS.2005.1532130.

  32. Dwyer T, Koren Y, Marriott K. Constrained graph layout by stress majorization and gradient projection. Discrete Mathematics, 2009, 309(7): 1895-1908. DOI: https://doi.org/10.1016/j.disc.2007.12.103.

    Article  MathSciNet  MATH  Google Scholar 

  33. Brin S, Page L. The anatomy of a large-scale hypertextual web search engine. Computer Networks and ISDN Systems, 1998, 30(1/2/3/4/5/6/7): 107-117. DOI: 10.1016/S0169-7552(98)00110-X.

  34. Davidson R, David H. Drawing graphs nicely using simulated annealing. ACM Transactions on Graphics, 1996, 15(4): 301-331. DOI: https://doi.org/10.1145/234535.234538.

    Article  Google Scholar 

  35. Gansner E R, Koren Y, North S. Graph drawing by stress majorization. In Proc. the 12th International Symposium on Graph Drawing, Sept. 2004, pp.239-250. DOI: 10.1007/978-3-540-31843-9_25.

  36. Purchase H C, Hoggan E, Görg C. How important is the “mental map"?|An empirical investigation of a dynamic graph layout algorithm. In Proc. the 14th International Symposium on Graph Drawing, Sept. 2006, pp.184-195. DOI: 10.1007/978-3-540-70904-6_19.

  37. Dwyer T. Scalable, versatile and simple constrained graph layout. Computer Graphics Forum, 2009, 28(3): 991-998. DOI: https://doi.org/10.1111/j.1467-8659.2009.01449.x.

    Article  Google Scholar 

  38. Newcomb T M. The Acquaintance Process. Holt, Rinehart & Winston, 1961. DOI: 10.1037/13156-000.

  39. Hristidis V, Hwang H, Papakonstantinou Y. Authoritybased keyword search in databases. ACM Transactions on Database Systems, 2008, 33(1): Article No. 1. DOI: 10.1145/1331904.1331905.

Download references

Author information

Authors and Affiliations

  1. ParisTech Elite Institute of Technology, Shanghai Jiao Tong University, Shanghai, 200240, China

    Shi-Ying Sheng & Chun-Yuan Wu

  2. Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    Sheng-Tao Chen & Xiao-Ju Dong

  3. Key Laboratory of Machine Perception (Ministry of Education), National Engineering Laboratory for Big Data Analysis and Application, Peking University, Beijing, 100080, China

    Xiao-Ru Yuan

Authors
  1. Shi-Ying Sheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sheng-Tao Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiao-Ju Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Chun-Yuan Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xiao-Ru Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiao-Ju Dong.

Supplementary Information

ESM 1

(PDF 411 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sheng, SY., Chen, ST., Dong, XJ. et al. Inverse Markov Process Based Constrained Dynamic Graph Layout. J. Comput. Sci. Technol. 36, 707–718 (2021). https://doi.org/10.1007/s11390-021-9910-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11390-021-9910-5

Keywords

Search

Navigation