Flow Complexity: Fast Polytopal Graph Complexity and 3D Object Clustering

  • Francisco Escolano
  • Daniela Giorgi
  • Edwin R. Hancock
  • Miguel A. Lozano
  • Bianca Falcidieno
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5534)


In this paper, we introduce a novel descriptor of graph complexity which can be computed in real time and has the same qualitative behavior of polytopal (Birkhoff) complexity, which has been successfully tested in the context of Bioinformatics. We also show how the phase-change point may be characterized in terms of the Laplacian spectrum, by analyzing the derivatives of the complexity function. In addition, the new complexity notion (flow complexity) is applied to cluster a database of Reeb graphs coming from analyzing 3D objects.


Heat Kernel Critical Area Geodesic Distance Graph Complexity Graph Permutation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Robles-Kelly, A., Hancock, E.R.: A riemannian approach to graph embedding. Pattern Recognition (40), 1042–1056 (2007)Google Scholar
  2. 2.
    Luo, B., Wilson, R.C., Hancock, E.: Spectral embedding of graphs. Pattern Recognition (36), 2213–2223 (2003)Google Scholar
  3. 3.
    Shokoufandeh, A., Dickinson, S., Siddiqi, K., Zucker, S.: Indexing using a spectral encoding of topological structure. In: IEEE ICPR, pp. 491–497Google Scholar
  4. 4.
    Torsello, A., Hancock, E.: Learning shape-classes using a mixture of tree-unions. IEEE Trans. on PAMI 28(6), 954–967 (2006)CrossRefGoogle Scholar
  5. 5.
    Lozano, M., Escolano, F.: Protein classification by matching and clustering surface graphs. Pattern Recognition 39(4), 539–551 (2006)CrossRefzbMATHGoogle Scholar
  6. 6.
    Escolano, F., Hancock, E., Lozano, M.: Birkhoff polytopes, heat kernels, and graph embedding. In: ICPR (2008)Google Scholar
  7. 7.
    Escolano, F., Hancock, E., Lozano, M.: Graph complexity, matrix permanents, and embedding. In: Proc. SSPR/SPR (2008)Google Scholar
  8. 8.
    Körner, J.: Coding of an information source having ambiguous alphabet and the entropy of graphs. In: Trans. of the 6th Prague Conference on Information Theory, pp. 411–425 (1973)Google Scholar
  9. 9.
    Estrada, E.: Graph spectra and structure in complex networks. Technical report, Institute of Complex Systems at Strathclyde, Department of Physics and Department of Mathematics, University of Strathclyde, Glasgow, UK (2008)Google Scholar
  10. 10.
    Qiu, H., Hancock, E.: Graph simplification and matching using conmute times. Pattern Recognition (40), 2874–2889 (2007)Google Scholar
  11. 11.
    López-Ruiz, R., Mancini, H., Calbet, X.: A statistical measure of complexity. Physics Letters A 209, 321–326 (1995)CrossRefGoogle Scholar
  12. 12.
    Reeb, G.: Sur les points singuliers d’une forme de Pfaff complètement intégrable ou d’une fonction numérique. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 222, 847–849 (1946)zbMATHGoogle Scholar
  13. 13.
    Shinagawa, Y., Kunii, T.L.: Constructing a Reeb Graph automatically from cross sections. IEEE Computer Graphics and Applications 11(6), 44–51 (1991)CrossRefGoogle Scholar
  14. 14.
    Biasotti, S.: Computational Topology Methods for Shape Modelling Applications. Ph.D thesis, Universitá degli Studi di Genova (May 2004)Google Scholar
  15. 15.
    Biasotti, S.: Topological coding of surfaces with boundary using Reeb graphs. Computer Graphics and Geometry 7(1), 31–45 (2005)Google Scholar
  16. 16.
    Hilaga, M., Shinagawa, Y., Kohmura, T., Kunii, T.L.: Topology matching for fully automatic similarity estimation of 3D shapes. In: SIGGRAPH 2001: Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, Los Angeles, CA, pp. 203–212. ACM Press, New York (2001)Google Scholar
  17. 17.
    Attene, M., Biasotti, S.: Shape retrieval contest 2008: Stability of watertight models. In: Spagnuolo, M., Cohen-Or, D., Gu, X. (eds.) SMI 2008: Proceedings IEEE International Conference on Shape Modeling and Applications, pp. 219–220 (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Francisco Escolano
    • 1
  • Daniela Giorgi
    • 2
  • Edwin R. Hancock
    • 3
  • Miguel A. Lozano
    • 1
  • Bianca Falcidieno
    • 2
  1. 1.Departamento de Ciencia de la Computación e Inteligencia ArtificialUniversity of AlicanteSpain
  2. 2.Istituto di Matematica Applicata e Tecnologie Informatiche Consiglio Nazionale delle RicercheItaly
  3. 3.Department of Computer ScienceUniversity of YorkUK

Personalised recommendations