Measuring Networks



We have adopted the view of graphs and, more generally, cell complexes as a domain upon which we may apply the tools of calculus to formulate differential equations and to analyze data. An important aspect of the discrete differential operators is that the operators are defined by the topology of the domain itself. Therefore, in an effort to provide a complete treatment of these differential operators, we examine in this chapter the properties of the network which may be extracted from the structure of these operators. In addition to the network properties extracted directly from the differential operators, we also review other methods for measuring the structural properties of a network. Specifically, the properties of the network that we consider are based on distances, partitioning, geometry, and topology. Our particular focus will be on the measurement of these properties from the graph structure. Applications will illustrate the use of these measures to predict the importance of nodes and to relate these measures to other properties of the subject being modeled by the network.


Cluster Coefficient Betti Number Laplacian Matrix Average Path Length Wiener Index 
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. 3.
    Albert, R., Jeong, H., Barabasi, A.: Diameter of the world wide web. Nature 401(6749), 130–131 (1999) CrossRefGoogle Scholar
  2. 6.
    Alon, N.: Eigenvalues and expanders. Combinatorica 6, 83–96 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 7.
    Alon, N., Milman, V.: λ 1, isoperimetric inequalities for graphs and superconcentrators. Journal of Combinatorial Theory. Series B 38, 73–88 (1985) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 16.
    Bader, D., Kintali, S., Madduri, K., Mihail, M.: Approximating betweenness centrality. In: Proc. of WAW, Lecture Notes in Computer Science, vol. 4863, pp. 124–137. Springer, Berlin (2007) Google Scholar
  5. 20.
    Barabasi, A., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999) MathSciNetCrossRefGoogle Scholar
  6. 34.
    Biggs, N.: Algebraic Graph Theory. Cambridge Tracts in Mathematics, vol. 67. Cambridge University Press, Cambridge (1974) zbMATHCrossRefGoogle Scholar
  7. 37.
    Biggs, N., Lloyd, E., Wilson, R.: Graph Theory, 1736–1936. Clarendon, Oxford (1986) zbMATHGoogle Scholar
  8. 42.
    Bonchev, D.: Information Theoretic Indices for Characterization of Chemical Structure. Research Studies Press, Chichester (1983) Google Scholar
  9. 43.
    Bonchev, D., Balaban, A., Liu, X., Klein, D.: Molecular cyclicity and centricity of polycyclic graphs. I. Cyclicity based on resistance distances or reciprocal distances. International Journal of Quantum Chemistry 50(1), 1–20 (1994) CrossRefGoogle Scholar
  10. 44.
    Bonchev, D., Rouvray, D.H. (eds.): Chemical Graph Theory—Introduction and Fundamentals. Gordon & Breach, New York (1991) zbMATHGoogle Scholar
  11. 45.
    Borgatti, S.: Identifying sets of key players in a social network. Computational and Mathematical Organization Theory 12(1), 21–34 (2006) zbMATHCrossRefGoogle Scholar
  12. 51.
    Bos̆njak, N., Mihalić, Z., Trinajstić, N.: Application of topographic indices to chromatographic data: Calculation of the retention indices of alkanes. Journal of Chromatography 540(1–2), 430–440 (1991) Google Scholar
  13. 57.
    Brandes, U.: A faster algorithm for betweenness centrality. Journal of Mathematical Sociology 25(2), 163–177 (2001) zbMATHCrossRefGoogle Scholar
  14. 74.
    Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Gunning, R. (ed.) Problems in Analysis, pp. 195–199. Princeton University Press, Princeton (1970) Google Scholar
  15. 81.
    Chung, F.R.K.: Spectral Graph Theory. Regional Conference Series in Mathematics, vol. 92. Am. Math. Soc., Providence (1997) zbMATHGoogle Scholar
  16. 87.
    Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms. MIT Press, Cambridge (2001) zbMATHGoogle Scholar
  17. 89.
    Costa, L., Rodrigues, F., Travieso, G., Boas, P.: Characterization of complex networks: A survey of measurements. Advances in Physics 56(1), 167–242 (2007) CrossRefGoogle Scholar
  18. 101.
    Demir, C., Gultekin, S., Yener, B.: Learning the topological properties of brain tumors. In: Proc. of IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB), vol. 2, pp. 262–270 (2005) Google Scholar
  19. 108.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Mathematik 1(1), 269–271 (1959) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 109.
    Diudea, M.V., Gutman, I.: Wiener-type topological indices. Croatica Chemica Acta 71(1), 21–51 (1998) Google Scholar
  21. 110.
    do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976) zbMATHGoogle Scholar
  22. 111.
    Dodziuk, J.: Difference equations, isoperimetric inequality and the transience of certain random walks. Transactions of the American Mathematical Society 284, 787–794 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 112.
    Dodziuk, J., Kendall, W.S.: Combinatorial Laplacians and isoperimetric inequality. In: Ellworthy, K.D. (ed.) From Local Times to Global Geometry, Control and Physics. Pitman Research Notes in Mathematics Series, vol. 150, pp. 68–74. Longman, Harlow (1986) Google Scholar
  24. 119.
    Edelsbrunner, H.: Geometry and Topology of Mesh Generation. Cambridge University Press, Cambridge (2001) CrossRefGoogle Scholar
  25. 124.
    Erdős, P., Rényi, A.: On random graphs. I. Publicationes Mathematicae Debrecen 6, 290–297 (1959) MathSciNetGoogle Scholar
  26. 125.
    Erdős, P., Rényi, A.: On the evolution of random graphs. Közlemények Publications 5, 17–61 (1960) Google Scholar
  27. 133.
    Fiedler, M.: Algebraic connectivity of graphs. Czechoslovak Mathematical Journal 23(98), 298–305 (1973) MathSciNetGoogle Scholar
  28. 143.
    Friedman, J.: Computing Betti numbers via combinatorial Laplacians. Algorithmica 21(4), 331–346 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 167.
    Grady, L., Schwartz, E.L.: Faster graph-theoretic image processing via small-world and quadtree topologies. In: Proc. of 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 2, pp. 360–365. IEEE Comput. Soc., Washington (2004) CrossRefGoogle Scholar
  30. 169.
    Grady, L., Schwartz, E.L.: Isoperimetric partitioning: A new algorithm for graph partitioning. SIAM Journal on Scientific Computing 27(6), 1844–1866 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 170.
    Graovac, A., Gutman, I., Trinajstić, N.: Topological Approach to the Chemistry of Conjugated Molecules. Springer, Berlin (1977) zbMATHCrossRefGoogle Scholar
  32. 171.
    Grassy, G., Calas, B., Yasri, A., Lahana, R., Woo, J., Iyer, S., Kaczorek, M., Floch, R., Buelow, R.: Computer-assisted rational design of immunosuppressive compounds. Nature Biotechnology 16, 748–752 (1998) CrossRefGoogle Scholar
  33. 175.
    Gremban, K., Miller, G., Zagha, M.: Performance evaluation of a new parallel preconditioner. In: Proceedings of 9th International Parallel Processing Symposium, pp. 65–69. IEEE Comput. Soc., Santa Barbara (1995) CrossRefGoogle Scholar
  34. 180.
    Gunduz, C., Yener, B., Gultekin, S.: The cell graphs of cancer. In: Proc. of International Symposium on Molecular Biology ISMB/ECCB, vol. 20, pp. 145–151 (2004) Google Scholar
  35. 181.
    Gutman, I., Kortvelyesi, T.: Wiener indices and molecular surfaces. Zeitschrift für Naturforschung A 50(7), 669–671 (1995) Google Scholar
  36. 182.
    Gutman, I., Mohar, B.: The quasi-Wiener and the Kirchhoff indices coincide. Journal of Chemical Information and Computer Sciences 36(5), 982–985 (1996) Google Scholar
  37. 187.
    Hansen, P., Jurs, P.: Chemical applications of graph theory: Part I. Fundamentals and topological indices. Journal of Chemical Education 65(7), 574–580 (1988) CrossRefGoogle Scholar
  38. 188.
    Harary, F.: Graph Theory. Addison-Wesley, Reading (1994) Google Scholar
  39. 196.
    Henle, M.: A Combinatorial Introduction to Topology. Dover, New York (1994) zbMATHGoogle Scholar
  40. 207.
    Hosoya, H.: A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons. Bulletin of the Chemical Society of Japan 44, 2332–2339 (1971) CrossRefGoogle Scholar
  41. 213.
    Ivanciuc, O., Taraviras, S.L., Cabrol-Bass, D.: Quasi-orthogonal basis set of molecular graph descriptors as a chemical diversity measure. Journal of Chemical Information and Computer Sciences 40, 126–134 (2000) Google Scholar
  42. 230.
    Kim, H., Kim, J.: Cyclic topology in complex networks. Physical Review E 72(3), 036109 (2005) CrossRefGoogle Scholar
  43. 235.
    Klein, D.J., Randić, M.: Resistance distance. Journal of Mathematical Chemistry 12(1), 81–95 (1993) MathSciNetCrossRefGoogle Scholar
  44. 247.
    Koschützki, D., Lehmann, K.A., Peeters, L., Richter, S., Tenfelde-Podehl, D., Zlotowski, O.: Centrality indices. In: Brandes, U., Erlebach, T. (eds.) Network Analysis. Lecture Notes in Computer Science, vol. 3418. Springer, Berlin (2005) Google Scholar
  45. 254.
    Lefschetz, S.: Algebraic Topology. Am. Math. Soc. Col. Pub., vol. 27. Am. Math. Soc., Providence (1942) zbMATHGoogle Scholar
  46. 265.
    Lukovits, I.: Decomposition of the Wiener topological index. Application to drug-receptor interactions. Journal of the Chemical Society. Perkin Transactions 2 9, 1667–1671 (1988) CrossRefGoogle Scholar
  47. 266.
    Lukovits, I.: Correlation between components of the Wiener index and partition coefficients of hydrocarbons. International Journal of Quantum Chemistry 44(s 19), 217–223 (1992) CrossRefGoogle Scholar
  48. 272.
    Massey, W.: A Basic Course in Algebraic Topology. Springer, Berlin (1993) Google Scholar
  49. 279.
    Merris, R.: An edge version of the matrix-tree theorem and the Wiener index. Linear and Multilinear Algebra 25(4), 291–296 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  50. 280.
    Meyer, M., Desbrun, M., Schröder, P., Barr, A.H.: Discrete differential-geometry operators for triangulated 2-manifolds. In: Hege, H.C., Polthier, K. (eds.) Visualization and Mathematics III, pp. 35–57. Springer, Heidelberg (2003) CrossRefGoogle Scholar
  51. 282.
    Mihalić, Z., Trinajstić, N.: A graph-theoretical approach to structure: Property relationships. Journal of Chemical Education 69(9), 701–712 (1992) CrossRefGoogle Scholar
  52. 283.
    Milgram, S.: The small world problem. Psychology Today 2(1), 60–67 (1967) MathSciNetGoogle Scholar
  53. 285.
    Mohar, B.: Isoperimetric inequalities, growth and the spectrum of graphs. Linear Algebra and Its Applications 103, 119–131 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  54. 286.
    Mohar, B.: Isoperimetric numbers of graphs. Journal of Combinatorial Theory. Series B 47, 274–291 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  55. 298.
    Opsahl, T., Panzarasa, P.: Clustering in weighted networks. Social Networks 31(2), 155–163 (2009) CrossRefGoogle Scholar
  56. 325.
    Rouvray, D.H.: The prediction of biological activity using molecular connectivity indices. Acta Pharmaceutica Jugoslavica 36, 239–251 (1986) Google Scholar
  57. 362.
    Strogatz, S.: Exploring complex networks. Nature 410(6825), 268–276 (2001) CrossRefGoogle Scholar
  58. 366.
    Surazhsky, T., Magid, E., Soldea, O., Elber, G., Rivlin, E.: A comparison of Gaussian and mean curvatures estimation methods on triangular meshes. In: Proc. of IEEE International Conference on Robotics and Automation ICRA’03, vol. 1 (2003) Google Scholar
  59. 367.
    Sylvester, J.J.: Chemistry and algebra. Nature 17(432), 284 (1878) CrossRefGoogle Scholar
  60. 370.
    Taraviras, S.L., Ivanciuc, O., Cabrol-Bass, D.: Identification of groupings of graph theoretical molecular descriptors using a hybrid cluster analysis approach. Journal of Chemical Information and Computer Sciences 40, 1128–1146 (2000) Google Scholar
  61. 396.
    Watts, D., Strogatz, S.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440–442 (1998) CrossRefGoogle Scholar
  62. 397.
    Watts, D.J.: Small Worlds: The Dynamics of Networks Between Order and Randomness. Princeton Studies in Complexity. Princeton University Press, Princeton (1999) Google Scholar
  63. 404.
    Wiener, H.: Correlation of heats of isomerization and differences in heats of vaporization of isomer among the paraffin hydrocarbons. Journal of the American Chemical Society 69, 2636–2638 (1947) CrossRefGoogle Scholar
  64. 405.
    Wiener, H.: Structural determination of paraffin boiling points. Journal of the American Chemical Society 69, 17–20 (1947) CrossRefGoogle Scholar
  65. 415.
    Zachary, W.W.: An information flow model for conflict and fission in small groups. Journal of Anthropological Research 33, 452–473 (1977) Google Scholar
  66. 425.
    Zomorodian, A.: Topology for Computing. Cambridge University Press, Cambridge (2005) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2010

Authors and Affiliations

  1. 1.Siemens Corporate ResearchPrincetonUSA
  2. 2.Athinoula A. Martinos Center for Biomedical Imaging, Department of RadiologyMassachusetts General Hospital, Harvard Medical SchoolCharlestownUSA

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