Simultaneous, Polynomial-Time Layout of Context Bigraph and Lattice Digraph

  • Tim PattisonEmail author
  • Aaron Ceglar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11511)


Formal Concept Analysis (FCA) takes as input the bipartite context graph and produces a directed acyclic graph representing the lattice of formal concepts. Excepting possibly the supremum and infimum, the set of formal concepts corresponds to the set of proper maximal bicliques in the context bigraph. This paper proposes polynomial-time graph layouts which emphasise maximal bicliques in the context bigraph and facilitate “reading” directed paths in the lattice digraph. These layouts are applied to sub-contexts of the InfoVis 2004 data set which are indivisible by the Carve divide-and-conquer FCA algorithm. The paper also investigates the relationship between vertex proximity in the bigraph layout and co-membership of maximal bicliques, and demonstrates the significant reduction of edge crossings in the digraph layout.


Lattice drawing Bigraph clustering Resistance distance 


  1. 1.
    Berry, A., Gutierrez, A., Huchard, M., Napoli, A., Sigayret, A.: Hermes: a simple and efficient algorithm for building the AOC-poset of a binary relation. Ann. Math. Artif. Intell. 72(1), 45–71 (2014). Scholar
  2. 2.
    Card, S., Mackinlay, J., Shneiderman, B.: Readings in Information Visualization: Using Vision to Think. Morgan Kaufmann, San Francisco (1999)Google Scholar
  3. 3.
    Cohen, J.D.: Drawing graphs to convey proximity: an incremental arrangement method. ACM Trans. Comput.-Hum. Interact. 4(3), 197–229 (1997). Scholar
  4. 4.
    Cox, T., Cox, M.: Multidimensional Scaling. Chapman Hall, London (1994)zbMATHGoogle Scholar
  5. 5.
    Cuffe, P., Keane, A.: Visualizing the electrical structure of power systems. IEEE Syst. J. 11(99), 1810–1821 (2017)CrossRefGoogle Scholar
  6. 6.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, New Jersey (1999)zbMATHGoogle Scholar
  7. 7.
    Didimo, W., Patrignani, M. (eds.): GD 2012. LNCS, vol. 7704. Springer, Heidelberg (2013). Scholar
  8. 8.
    Doerfel, S., Jäschke, R., Stumme, G.: Publication analysis of the formal concept analysis community. In: Domenach, F., Ignatov, D.I., Poelmans, J. (eds.) ICFCA 2012. LNCS (LNAI), vol. 7278, pp. 77–95. Springer, Heidelberg (2012). Scholar
  9. 9.
    Doyle, P., Snell, J.: Random Walks and Electric Networks. The Mathematical Association of America (1984)Google Scholar
  10. 10.
    Eades, P., Wormald, N.C.: Edge crossings in drawings of bipartite graphs. Algorithmica 11, 379–403 (1994)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Estrada, E., Hatano, N.: Resistance distance, information centrality, node vulnerability and vibrations in complex networks. In: Estrada, E., Fox, M., Higham, D.J., Oppo, G.L. (eds.) Network Science: Complexity in Nature and Technology, pp. 13–29. Springer, London (2010). Scholar
  12. 12.
    Freese, R.: Automated lattice drawing. In: Eklund, P. (ed.) ICFCA 2004. LNCS (LNAI), vol. 2961, pp. 112–127. Springer, Heidelberg (2004). Scholar
  13. 13.
    Ganter, B., Wille, R.: Formal Concept Analysis. Springer, Heidelberg (1999). Scholar
  14. 14.
    Ganter, B.: Conflict avoidance in additive order diagrams. J. Univers. Comput. Sci. 10(8), 955–966 (2004)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gervacio, S.V.: Resistance distance in complete n-partite graphs. Discrete Appl. Math. 203 (2016). Scholar
  16. 16.
    Gibson, H., Faith, J., Vickers, P.: A survey of two-dimensional graph layout techniques for information visualisation. Inf. Vis. 12(3–4), 324–357 (2012)Google Scholar
  17. 17.
    Gross, J.L., Yellen, J., Zhang, P. (eds.): Discrete Mathematics and its Applications. Handbook of Graph Theory, 2nd edn. Chapman and Hall/CRC Press, New York (2013)Google Scholar
  18. 18.
    Gutman, I., Xiao, W.: Generalized inverse of the Laplacian matrix and some applications. Bulletin: Classe des sciences mathématiques et naturelles 129(29), 15–23 (2004). Scholar
  19. 19.
    Hannan, T., Pogel, A.: Spring-based lattice drawing highlighting conceptual similarity. In: Missaoui, R., Schmidt, J. (eds.) ICFCA 2006. LNCS (LNAI), vol. 3874, pp. 264–279. Springer, Heidelberg (2006). Scholar
  20. 20.
    Ho, N.D., van Dooren, P.: On the pseudo-inverse of the Laplacian of a bipartite graph. Appl. Math. Lett. 18, 917–922 (2005)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Klein, D., Randić, M.: Resistance distance. J. Math. Chem. 12, 81–85 (1993)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Klimenta, M., Brandes, U.: Graph drawing by classical multidimensional scaling: new perspectives. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 55–66. Springer, Heidelberg (2013). Scholar
  23. 23.
    Kunegis, J.: Exploiting the structure of bipartite graphs for algebraic and spectral graph theory applications. Internet Math. 11(3), 201–321 (2015). Scholar
  24. 24.
    Kunegis, J., Schmidt, S., Albayrak, Ş., Bauckhage, C., Mehlitz, M.: Modeling collaborative similarity with the signed resistance distance kernel. In: Ghallab, M. (ed.) European Conference on Artificial Intelligence. IOS Press (2008)Google Scholar
  25. 25.
    Kuznetsov, S.O., Poelmans, J.: Knowledge representation and processing with Formal Concept Analysis. Wiley Interdisc. Rev. Data Min. Know. Disc. 3(3), 200–215 (2013). Scholar
  26. 26.
    von Luxburg, U., Radl, A., Hein, M.: Getting lost in space: large sample analysis of the resistance distance. In: Advances in Neural Information Processing Systems 23: 24th Annual Conference on Neural Information Processing Systems 2010. pp. 2622–2630. Curran, New York (2010)Google Scholar
  27. 27.
    Pattison, T., Ceglar, A.: Interaction challenges for the dynamic construction of partially-ordered sets. In: Bertet, K., Rudolph, S. (eds.) Proceedings of 11th International Conference on Concept Lattices and their Applications, pp. 23–34. CEUR Workshop Proceedings, Košice, Slovakia (2014).
  28. 28.
    Pattison, T., Ceglar, A., Weber, D.: Efficient Formal Concept Analysis through recursive context partitioning. In: Ignatov, D.I., Nourine, L. (eds.) Proceedings of 14th International Conference on Concept Lattices & Their Applications, vol. 2123. CEUR Workshop Proceedings. Czech Republic (2018).
  29. 29.
    Pattison, T., Weber, D., Ceglar, A.: Enhancing layout and interaction in Formal Concept Analysis. In: Proceedings of 2014 IEEE Pacific Visualization Symposium (PacificVis), pp. 248–252 (2014).
  30. 30.
    Plaisant, C., Fekete, J.D., Grinstein, G.: Promoting insight-based evaluation of visualizations: from contest to benchmark repository. IEEE Trans. Vis. Comput. Graph. 14(1), 120–134 (2008). Scholar
  31. 31.
    Pohlmann, J.: Configurable Graph Drawing Algorithms for the TikZ Graphics Description Language. Master’s thesis, Univerisität zu Lübeck (2011).
  32. 32.
    Rival, I.: Reading, drawing, and order. In: Rosenberg, I.G., Sabidussi, G. (eds.) Algebras and Orders, vol. 389, pp. 359–404. Springer, Dordrecht (1993). Scholar
  33. 33.
    Stephenson, K., Zelen, M.: Rethinking centrality: methods and applications. Soc. Netw. 11, 1–37 (1989)CrossRefGoogle Scholar
  34. 34.
    Sugiyama, K., Tagawa, S., Toda, M.: Methods for visual understanding of hierarchical system structures. IEEE Trans. Syst. Man Cybern. 11(2), 109–125 (1981)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Tantau, T.: Graph drawing in TikZ. In: Didimo and Patrignani [7], pp. 517–528.
  36. 36.
    Wilkinson, J.: The Algebraic Eigenvalue Problem. Oxford University Press, Oxford (1965)zbMATHGoogle Scholar
  37. 37.
    Zschalig, C.: An FDP-algorithm for drawing lattices. In: Diatta, J., Eklund, P., Liquire, M. (eds.) Proceedings of CLA 2007, vol. 331, pp. 58–71. (2007).

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Authors and Affiliations

  1. 1.Defence Science and Technology GroupAdelaideAustralia

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