Review of Graph Invariants for Quantitative Analysis of Structure Dynamics

  • Wojciech Czech
  • Witold Dzwinel
Part of the Studies in Computational Intelligence book series (SCI, volume 416)


In this work we review graph invariants used for quantitative analysis of evolving graphs. Focusing on graph datasets derived from structural pattern recognition and complex networks fields, we demonstrate how to capture relevant topological features of networks. In an experimental setup, we study structural properties of graphs representing rotating 3D objects and show how they are related to characteritics of undelying images. We present how evolving strucure of Autonomous Systems (ASs) network is reflected by non-trivial changes in scalar graph descriptors. We also inspect characteristics of growing tumor vascular networks, obtained from a simulation. Additionally, the overview of currently used graph invariants with several possible groupings is provided.


Graphic Processing Unit Cellular Automaton Heat Kernel Vascular Network Betweenness Centrality 
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.
    Iapr technical commitee 15 (January 31, 2012),
  2. 2.
    Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Reviews of modern physics 74(1), 47 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Baish, J.W., Jain, R.K.: Fractals and cancer. Cancer Research 60(14), 3683 (2000)Google Scholar
  4. 4.
    Barabási, A.L., Oltvai, Z.N.: Network biology: understanding the cell’s functional organization. Nature Reviews Genetics 5(2), 101–113 (2004)CrossRefGoogle Scholar
  5. 5.
    Barrat, A., Barthlemy, M., Vespignani, A.: Dynamical processes on complex networks. Cambridge University Press (2008)Google Scholar
  6. 6.
    Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.U.: Complex networks: Structure and dynamics. Physics Reports 424(4-5), 175–308 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Buluç, A., Gilbert, J.R., Budak, C.: Solving path problems on the gpu. Parallel Computing 36(5-6), 241–253 (2010)zbMATHCrossRefGoogle Scholar
  8. 8.
    Carmeliet, P., Jain, R.K.: Angiogenesis in cancer and other diseases. NATURE-LONDON, 249–257 (2000)Google Scholar
  9. 9.
    Chung, F.R.K.: Spectral Graph Theory. CBMS Regional Conference Series in Mathematics, vol. 92, p. 3, 8. American Mathematical Society (1997)Google Scholar
  10. 10.
    Conte, D., Foggia, P., Sansone, C., Vento, M.: Thirty years of graph matching in pattern recognition. International Journal of Pattern Recognition and Artificial Intelligence 18(3), 265–298 (2004)CrossRefGoogle Scholar
  11. 11.
    Costa, L.F., Rodrigues, F.A., Travieso, G., Boas, P.: Characterization of complex networks: A survey of measurements. Advances in Physics 56(1), 167–242 (2007)CrossRefGoogle Scholar
  12. 12.
    Czech, W.: Graph descriptors from b-matrix representation. In: Jiang, X., Ferrer, M., Torsello, A. (eds.) GbRPR 2011. LNCS, vol. 6658, pp. 12–21. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  13. 13.
    Czech, W., Goryczka, S., Arodz, T., Dzwinel, W., Dudek, A.: Exploring complex networks with graph investigator research application. Computing and Informatics 30(2) (2011)Google Scholar
  14. 14.
    Czech, W., Yuen, D.A.: Efficient graph comparison and visualization using gpu. In: Proceedings of the 14th IEEE International Conference on Computational Science and Engineering (CSE 2011), pp. 561–566 (2011), doi:10.1109/CSE.2011.223Google Scholar
  15. 15.
    Estrada, E.: Generalized walks-based centrality measures for complex biological networks. Journal of Theoretical Biology 263(4), 556–565 (2010)CrossRefGoogle Scholar
  16. 16.
    Estrada, E., Fox, M., Higham, D., Oppo, G.: Network Science. Springer (2010)Google Scholar
  17. 17.
    Estrada, E., Higham, D.: Network properties revealed through matrix functions. SIAM Review 52(4), 696–714 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Estrada, E., Higham, D., Hatano, N.: Communicability betweenness in complex networks. Physica A: Statistical Mechanics and its Applications 388(5), 764–774 (2009)CrossRefGoogle Scholar
  19. 19.
    Freeman, L.: A set of measures of centrality based on betweenness. Sociometry, 35–41 (1977)Google Scholar
  20. 20.
    Freeman, L.: Centrality in social networks conceptual clarification. Social Networks 1(3), 215–239 (1979)CrossRefGoogle Scholar
  21. 21.
    Gazit, Y., Berk, D., Leunig, M., Baxter, L., Jain, R.: Scale-invariant behavior and vascular network formation in normal and tumor tissue. Physical Review Letters 75(12), 2428–2431 (1995)CrossRefGoogle Scholar
  22. 22.
    Gfeller, D., Rios, P.D.L., Caflisch, A., Rao, F.: Complex network analysis of free-energy landscapes. Proceedings of the National Academy of Sciences 104(6), 1817 (2007)CrossRefGoogle Scholar
  23. 23.
    Godsil, C., Royle, G.: Algebraic graph theory. Springer, New York (2001)zbMATHCrossRefGoogle Scholar
  24. 24.
    Gordon, M., Scantlebury, G.: Non-random polycondensation: Statistical theory of the substitution effect. Transactions of the Faraday Society 60, 604–621 (1964)CrossRefGoogle Scholar
  25. 25.
    Harris, C., Stephens, M.: A combined corner and edge detector. In: Alvey Vision Conference, Manchester, UK, vol. 15, p. 50 (1988)Google Scholar
  26. 26.
    Jamakovic, A., Uhlig, S.: On the relationships between topological measures in real-world networks. Networks and Heterogeneous Media 3(2), 345 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Képès, F.: Biological networks. World Scientific Pub. Co. Inc. (2007)Google Scholar
  28. 28.
    Leskovec, J.: Stanford large network dataset collection,
  29. 29.
    Leskovec, J., Kleinberg, J., Faloutsos, C.: Graph evolution: Densification and shrinking diameters. ACM Transactions on Knowledge Discovery from Data (TKDD) 1(1), 2–es (2007)Google Scholar
  30. 30.
    Luo, B., Wilson, R., Hancock, E.: Spectral embedding of graphs. Pattern Recognition 36(10), 2213–2230 (2003)zbMATHCrossRefGoogle Scholar
  31. 31.
    Luo, Y., Lin, H., Huang, M., Liaw, T.: Conformation-networks of two-dimensional lattice homopolymers. Physics Letters A 359(3), 211–217 (2006)CrossRefGoogle Scholar
  32. 32.
    Marchette, D.: Random graphs for statistical pattern recognition. Wiley-IEEE (2004)Google Scholar
  33. 33.
    Marfil, R., Escolano, F., Bandera, A.: Graph-Based Representations in Pattern Recognition and Computational Intelligence. In: Cabestany, J., Sandoval, F., Prieto, A., Corchado, J.M. (eds.) IWANN 2009. LNCS, vol. 5517, pp. 399–406. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  34. 34.
    Nene, S., Nayar, S., Murase, H.: Columbia object image library (coil-20). Dept. Comput. Sci., Columbia Univ., New York (1996),
  35. 35.
    Newman, M.: A measure of betweenness centrality based on random walks. Social networks 27(1), 39–54 (2005)CrossRefGoogle Scholar
  36. 36.
    Newman, M.: The mathematics of networks. The New Palgrave Encyclopedia of Economics (2007)Google Scholar
  37. 37.
    Page, L., Brin, S., Motwani, R., Winograd, T.: The pagerank citation ranking: Bringing order to the web (1999)Google Scholar
  38. 38.
    Platt, J.: Influence of neighbor bonds on additive bond properties in paraffins. The Journal of Chemical Physics 15, 419 (1947)CrossRefGoogle Scholar
  39. 39.
    Qiu, H., Hancock, E.: Clustering and embedding using commute times. IEEE Transactions on Pattern Analysis and Machine Intelligence 29(11), 1873–1890 (2007)CrossRefGoogle Scholar
  40. 40.
    Song, C., Havlin, S., Makse, H.: Self-similarity of complex networks. Nature 433 (2005)Google Scholar
  41. 41.
    Topa, P.: Dynamically reorganising vascular networks modelled using cellular automata approach. Cellular Automata, 494–499 (2010)Google Scholar
  42. 42.
    Wasserman, S., Faust, K.: Social network analysis: Methods and applications. Cambridge Univ. Pr. (1994)Google Scholar
  43. 43.
    Wcisło, R., Dzwinel, W., Gosztyla, P., Yuen, D.A., Czech, W.: Interactive visualization tool for planning cancer treatment. Tech. rep., University of Minnesota Supercomputing Institute Research Report UMSI 2011/7, CB number 2011-4 (2011)Google Scholar
  44. 44.
    Wcisło, R., Dzwinel, W., Yuen, D., Dudek, A.: A 3-d model of tumor progression based on complex automata driven by particle dynamics. Journal of Molecular Modeling 15(12), 1517–1539 (2009)CrossRefGoogle Scholar
  45. 45.
    Welter, M., Rieger, H.: Physical determinants of vascular network remodeling during tumor growth. Eur. Phys. J. E 33, 149–163 (2010)CrossRefGoogle Scholar
  46. 46.
    Wiener, H.: Structural determination of paraffin boiling points. Journal of the American Chemical Society 69(1), 17–20 (1947)CrossRefGoogle Scholar
  47. 47.
    Wilson, R., Hancock, E., Luo, B.: Pattern vectors from algebraic graph theory. IEEE Transactions on Pattern Analysis and Machine Intelligence 27(7), 1112–1124 (2005)CrossRefGoogle Scholar
  48. 48.
    Xiao, B., Hancock, E., Wilson, R.: A generative model for graph matching and embedding. Computer Vision and Image Understanding 113(7), 777–789 (2009)CrossRefGoogle Scholar
  49. 49.
    Xiao, B., Hancock, E., Wilson, R.: Graph characteristics from the heat kernel trace. Pattern Recognition 42(11), 2589–2606 (2009)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.AGH University of Science and TechnologyKrakówPoland

Personalised recommendations