In this paper we aim to characterize graphs in terms of structural complexities. Our idea is to decompose a graph into substructures of increasing layers, and then to measure the dissimilarity of these substructures using Jensen-Shannon divergence. We commence by identifying a centroid vertex by computing the minimum variance of its shortest path lengths. From the centroid vertex, a family of centroid expansion subgraphs of the graph with increasing layers are constructed. We then compute the depth-based complexity trace of a graph by measuring how the Jensen-Shannon divergence varies with increasing layers of the subgraphs. The required Shannon or von Neumann entropies are computed on the condensed subgraph family of the graph. We perform graph clustering in the principal components space of the complexity trace vector. Experiments on graph datasets abstracted from bioinformatics and image data demonstrate effectiveness and efficiency of the graphs complexity traces.


Shannon Entropy Short Path Length Graph Complexity Graph Kernel Graph Edit Distance 
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.


  1. 1.
    Crutchfield, J.P., Shalizi, C.R.: Thermodynamic depth of causal states: Objective complexity via minimal representations. Physical Review E 59, 275–283 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Escolano, F., Hancock, E.R., Lozano, M.A.: Heat diffusion: Thermodynamic depth complexity of networks. Physical Review E 85, 206–236 (2012)CrossRefGoogle Scholar
  3. 3.
    Han, L., Escolano, F., Hancock, E.R.: Graph characterizations from von Neumann entropy. To appear in Pattern Recognition Letter (2012)Google Scholar
  4. 4.
    Martins, A.F.T., Smith, N.A., Xing, E.P., Aguiar, P.M.Q., Figueiredo, M.A.T.: Nonextensive information theoretic kernels on measures. Journal of Machine Learning Research 10, 935–975 (2009)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bunke, H., Riesen, K.: Improving vector space embedding of graphs through feature selection algorithms. Pattern Recognition 44, 1928–1940 (2011)CrossRefGoogle Scholar
  6. 6.
    Bai, L., Hancock, E.R.: Graph Clustering Using the Jensen-Shannon Kernel. In: Real, P., Diaz-Pernil, D., Molina-Abril, H., Berciano, A., Kropatsch, W. (eds.) CAIP 2011, Part I. LNCS, vol. 6854, pp. 394–401. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Ren, P., Wilson, R.C., Hancock, E.R.: Graph Characterization via Ihara Coefficients. IEEE Transactions on Neural Networks 22, 233–245 (2011)CrossRefGoogle Scholar
  8. 8.
    Dehmer, M., Mowshowitz, A.: A history of graph entropy measures. Proceedings of Information Sciences 181, 57–78 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Shervashidze, N., Schweitzer, P., Leeuwen, E.J., Mehlhorn, K., Borgwardt, K.M.: Weisfeiler-Lehman graph kernels. Journal of Machine Learning Research 1, 1–48 (2010)Google Scholar
  10. 10.
    Wilson, R.C., Hancock, E.R., Luo, B.: Pattern vectors from algebraic graph theory. IEEE Transactions Pattern Analysis and Machine Intelligence 27, 1112–1124 (2005)CrossRefGoogle Scholar
  11. 11.
    Han, L., Hancock, E.R., Wilson, R.C.: Learning Generative Graph Prototypes Using Simplified von Neumann Entropy. In: Jiang, X., Ferrer, M., Torsello, A. (eds.) GbRPR 2011. LNCS, vol. 6658, pp. 42–51. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. 12.
    Passerini, F., Severini, S.: Quantifying complexity in networks: the von Neumann entropy. International Journal of Agent Technologies and Systems 1, 58–67 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Lu Bai
    • 1
  • Edwin R. Hancock
    • 1
  1. 1.Department of Computer ScienceUniversity of York, UKHeslingtonUK

Personalised recommendations