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Spectral Characterization of Network Structures and Dynamics

  • Anirban BanerjeeEmail author
  • Jürgen JostEmail author
Chapter
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

Mathematically, graphs defy a systematic and complete classification, and empirically, the graphs representing networks come in a bewildering multitude. We have developed some tools (8; 9; 710) that at least allow for a rough classification of graphs that reflects the difference in the empirical domains from which network data are produced and that does not depend on sophisticated visualization tools.

As such, a graph is a rather simple formal structure. It consists of nodes or vertices that are connected by edges or links. These nodes then represent the elements of a network (and we shall often not distinguish between the network and its underlying graph), and the edges represent relations between them. These could be chemical interactions as in intracellular networks of genes, proteins, or metabolites, synaptic connections between neurons, physical links in infrastructural networks, links between Internet pages, co-occurrences between words in sentences or on text pages, email contacts between people, co-authorships between scientists, and so on. This structure then can be expected to be somehow adapted to the function of the network, by evolution, self-organization, or design. In turn, any dynamics supported by the network will be constrained by this underlying structure.

Our approach is based on associating certain mathematical objects—which ultimately just yield some numbers—to a graph which reflect its structural properties and which in particular encode the constraints on the dynamics that it can support. The mathematical objects will be an operator, the graph Laplacian (a discrete analogue of the Laplace operator in real analysis), and its eigenfunctions, and the numbers alluded to will be the eigenvalues of that operator.

References

  1. 1.
    L.A. Adamic and N. Glance, The political blogosphere and the 2004 US election: Divided they blog, in Proceedings of the WWW-2005 Workshop on the Weblogging Ecosystem (2005)Google Scholar
  2. 2.
    R. Albert, A.-L. Barabási, Statistical mechanics of complex networks, Reviews of Modern Physics 74, 2002, 47–97CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    F.M. Atay, T. Biyikoğlu, J. Jost, Synchronization of networks with prescribed degree distributions, IEEE Trans. Circuits and Systems I 53 (1), 2006, 92–98Google Scholar
  4. 4.
    F.M. Atay, T. Biyikoğlu, J. Jost, Network synchronization: Spectral versus statistical properties, Phys. D 224, 2006, 35–41CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    F.M. Atay, J. Jost, A. Wende, Delays, connection topology, and synchronization of coupled chaotic maps, Phys. Rev. Lett. 92 (14), 2004, 144101CrossRefGoogle Scholar
  6. 6.
    A. Banerjee, J. Jost, Laplacian spectrum and protein-protein interaction networks, preprintGoogle Scholar
  7. 7.
    A. Banerjee, J. Jost, On the spectrum of the normalized graph Laplacian, Lin. Alg. Appl. 428, 2008, 3015–3022CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    A. Banerjee, J. Jost, Graph spectra as a systematic tool in computational biology, Discr. Appl. Math., to appearGoogle Scholar
  9. 9.
    A. Banerjee, J. Jost, Spectral plots and the representation and interpretation of biological data, Theory Biosc. 126, 2007, 15–21CrossRefGoogle Scholar
  10. 10.
    A. Banerjee, J. Jost, Spectral plot properties: Towards a qualitative classification of networks, NHM 3, 2008, 395–411CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    A.-L. Barabási, R.A. Albert, Emergence of scaling in random networks, Science 286, 1999, 509–512CrossRefMathSciNetGoogle Scholar
  12. 12.
    P. Blanchard, T. Krüger, The “Cameo” principle and the origin of scale-free graphs in social networks, J. Stat. Phys. 114, 1399–1416, 2004CrossRefzbMATHGoogle Scholar
  13. 13.
    T. Biyikoğlu, J. Leydold, P. Stadler, Laplacian Eigenvectors of Graphs, Springer Berlin, 2007zbMATHGoogle Scholar
  14. 14.
    B. Bolobás, Modern Graph Theory, Springer, Berlin, 1998Google Scholar
  15. 15.
    F. Chung, Spectral Graph Theory, AMS, Providence, RI, 1997Google Scholar
  16. 16.
    F. Chung, L. Y. Lu, Complex Graphs and Networks, AMS, Providence, RI, 2006Google Scholar
  17. 17.
    S.N. Dorogovtsev, J.F.F. Mendes, Evolution of Networks, Oxford University Press, Oxford, 2003.CrossRefzbMATHGoogle Scholar
  18. 18.
    M. Faloutsos et al., On power-law relationships of the Internet topology, SIG-COMM, 1999.Google Scholar
  19. 19.
    P.M. Gleiser, L. Danon, Community structure in Jazz, Advances in Complex Systems (ACS) 6 (4), 2003, 565–573CrossRefGoogle Scholar
  20. 20.
    C. Godsil, G. Royle, Algebraic Graph Theory, Springer, Berlin, 2001zbMATHGoogle Scholar
  21. 21.
    R. Guimera et al, Self-similar community structure in a network of human interactions, Physical Review E 68, 2003, 065103(R)CrossRefGoogle Scholar
  22. 22.
    M. Horton, H. Stark, A. Terras, What are zeta functions of graphs and what are they good for? In Quantum graphs and their applications, Contemp. Math., Amer. Math. Soc., Providence, RI, 415, 2006, 173–189Google Scholar
  23. 23.
    M. Ipsen, A.S. Mikhailov, Evolutionary reconstruction of networks, Phys. Rev. E 66(4), 046109, 2002Google Scholar
  24. 24.
    H. Jeong et al, The large-scale organization of metabolic networks, Nature 407, 2000, 651–654CrossRefGoogle Scholar
  25. 25.
    J. Jost, Mathematical methods in biology and neurobiology, monograph, to appearGoogle Scholar
  26. 26.
    J. Jost, in: J.F. Feng, J. Jost, M.P. Qian (eds.), Networks: From Biology to Theory, 35–62, Springer, Berlin, 2007CrossRefzbMATHGoogle Scholar
  27. 27.
    J. Jost, M.P. Joy, Spectral properties and synchronization in coupled map lattices, Phys. Rev. E 65 (1), 2002, 016201CrossRefMathSciNetGoogle Scholar
  28. 28.
    J. Jost, M.P. Joy, Evolving networks with distance preferences, Phys. Rev. E 66, 2002, 36126–36132CrossRefMathSciNetGoogle Scholar
  29. 29.
    D.H. Kim, A. Motter, Ensemble averageability in network spectra, Phys. Rev. Lett. 98, 2007, 248701CrossRefGoogle Scholar
  30. 30.
    J. Kleinberg et al, The Web as a Graph: Measurements, Models, and Methods, LNCS 1627, 1999, 1–17MathSciNetGoogle Scholar
  31. 31.
    P. Krapivsky, S. Redner, Network growth by copying, Phys. Rev. E 71, 2005, 036118CrossRefMathSciNetGoogle Scholar
  32. 32.
    R. Merris, Laplacian matrices of graphs – A survey, Lin. Alg. Appl. 198, 1994, 143–176CrossRefMathSciNetGoogle Scholar
  33. 33.
    R Milo et al, Network motifs: Simple building blocks of complex networks, Science 298, 2002, 824–827CrossRefGoogle Scholar
  34. 34.
    R. Milo et al, Superfamilies of evolved and designed networks, Science 303, 2004, 1538–1542CrossRefGoogle Scholar
  35. 35.
    B. Mohar, Some applications of Laplace eigenvalues of graphs, in: G. Hahn, G. Sabidussi (eds.), Graph Symmetry: Algebraic Methods and Applications, 227–277, Springer, Berlin, 1997Google Scholar
  36. 36.
    R. Monasson, Diffusion, localization and dispersion relations on “small-world” lattices, Europ. Phys. J. B 12, 1999, 555–567CrossRefGoogle Scholar
  37. 37.
    M.E.J. Newman, The structure of scientific collaboration networks, Proc. Natl. Acad. Sci. USA 98, 2001, 404–409CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    M.E.J. Newman, Finding community structure in networks using the eigenvectors of matrices, Phys. Rev. E 74, 2006, 036104CrossRefMathSciNetGoogle Scholar
  39. 39.
    M. Newman, The structure and function of complex networks, SIAM Review 45, 2003, 167–256CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    S. Ohno, Evolution by Gene Duplication, Springer, Berlin, 1970Google Scholar
  41. 41.
    L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64, 1990, 821–824CrossRefMathSciNetGoogle Scholar
  42. 42.
    A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization – A Universal Concept in Nonlinear Science, Cambridge University Press, Cambridge, 2001CrossRefGoogle Scholar
  43. 43.
    G. Rangarajan, M.Z. Ding, Stability of synchronized chaos in coupled dynamical systems, Phys. Lett. A 296, 2002, 204–212CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    H. Simon, On a class of skew distribution functions, Biometrika 42, 1955, 425–440zbMATHMathSciNetGoogle Scholar
  45. 45.
    R. Solé et al, A model of large scale proteome evolution, Adv. Compl. Syst. 5, 2002, 43–54CrossRefzbMATHGoogle Scholar
  46. 46.
    A. Vazquez et al., Modelling of protein interaction networks, ComPlexUs 1, 2003, 38–44Google Scholar
  47. 47.
    A. Wagner, How the global structure of protein interaction networks evolves, Proc. Roy. Soc. B 270, 2003, 457–466CrossRefGoogle Scholar
  48. 48.
    A. Wagner, Evolution of gene networks by gene duplications — A mathematical model and its implications on genome organization, Proc. Nat. Acad. Sciences USA 91 (10), 1994, 4387–4391CrossRefzbMATHGoogle Scholar
  49. 49.
    D.J. Watts, S.H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature 393, 1998, 440–442CrossRefGoogle Scholar
  50. 50.
    J.G. White et al, The structure of the nervous system of the nematode Caenorhabditis elegans, Phil. Trans. Royal Soc. of London Series B-Bio. Sc. 314, 1986, 1–340CrossRefGoogle Scholar
  51. 51.
    P. Zhu, R. Wilson, A study of graph spectra for comparing graphs. In Proc. of British Machine Vision Conf. (MBVC), Sep 2005Google Scholar
  52. 52.
    K.H. Wolfe, D.C. Shields, Molecular evidence for an ancient duplication of the entire yeast genome, Nature 387 (6634), 1997, 708–713CrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Max Planck Institute for Molecular GeneticsIhnestr. 63-73BerlinGermany
  2. 2.Max Planck Institute for Mathematics in the Sciences, Inselstr.22, 04103 Leipzig, Germanyand Santa Fe Institute, Santa FeNMUSA

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