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Polytopal Graph Complexity, Matrix Permanents, and Embedding

  • Francisco Escolano
  • Edwin R. Hancock
  • Miguel A. Lozano
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5342)

Abstract

In this paper, we show how to quantify graph complexity in terms of the normalized entropies of convex Birkhoff combinations. We commence by demonstrating how the heat kernel of a graph can be decomposed in terms of Birkhoff polytopes. Drawing on the work of Birkhoff and von Neuman, we next show how to characterise the complexity of the heat kernel. Finally, we provide connections with the permanent of a matrix, and in particular those that are doubly stochastic. We also include graph embedding experiments based on polytopal complexity, mainly in the context of Bioinformatics (like the clustering of protein-protein interaction networks) and image-based planar graphs.

Keywords

Bipartite Graph Heat Kernel Permutation Matrice Matrix Permanent Graph Complexity 
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.

References

  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.
    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
  7. 7.
    Escolano, F., Hancock, E., Lozano, M.: Birkhoff polytopes, heat kernels, and graph embedding. In: ICPR (2008)Google Scholar
  8. 8.
    Birkhoff, G.D.: Tres observaciones sobre el algebra lineal. Universidad Nacional de Tucuman Revista, Serie A 5, 147–151 (1946)Google Scholar
  9. 9.
    Chang, C., Chen, W., Huang, H.: On service guarangees for input buffered crossbar switches: A capacity decomposition approach by birkhoff and von neumann. In: IEEE IWQoS, pp. 79–86 (1998)Google Scholar
  10. 10.
    Kondor, R.I., Lafferty, J.: Diffusion kernels on graphs and other discrete structures. In: Proc. ICML (2002)Google Scholar
  11. 11.
    Mirsky, L.: Proofs of two theorems on doubly stochastic matrices. Proc. Amer. Math. Soc. 9, 371–374 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. Journal of the ACM 51(4), 671–697 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Qiu, H., Hancock, E.: Graph simplification and matching using conmute times. Pattern Recognition (40), 2874–2889 (2007)Google Scholar
  14. 14.
    von Mering, C., Huynen, M., Jaeggi, D., Schmidt, S., Bork, P., Snell, B.: String: a database of predicted functional associations. Nuc. Acid Res. 31, 258–261 (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Francisco Escolano
    • 1
  • Edwin R. Hancock
    • 2
  • Miguel A. Lozano
    • 1
  1. 1.Departamento de Ciencia de la Computación e Inteligencia ArtificialUniversity of AlicanteSpain
  2. 2.Department of Computer ScienceUniversity of YorkUK

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