Complex Networks

  • Nicolás RubidoEmail author
Part of the Springer Theses book series (Springer Theses)


This chapter contains known definitions and (a few novel) results from Network Theory. This thesis is based on the mathematical background, the concepts, the network characterisation methods, and the results we present here. Hence, this chapter constitutes the methods we use to address the problem of the transmission of energy and synchronisation in complex networks. Specifically, our results on the transmission of energy are based on the knowledge and understanding of the network structures where the energy is transmitted and our results on synchronisation are mainly based on the topological and functional way the dynamical units are interconnected to form a complex network.


  1. 1.
    E.J. Henley, R.A. Williams, Graph Theory in Modern Engineering (Academic Press, New York, 1973)zbMATHGoogle Scholar
  2. 2.
    N. Biggs, Algebraic Graph Theory, 2nd edn. (Cambridge University Press, New York, 1974)CrossRefzbMATHGoogle Scholar
  3. 3.
    F.R.K. Chung, Spectral Graph Theory (American Mathematical Society, Providence, 1997)zbMATHGoogle Scholar
  4. 4.
    B. Bollobás, Modern Graph Theory (Springer, New York, 1998)CrossRefzbMATHGoogle Scholar
  5. 5.
    R. Albert, A.L. Barabási, Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–98 (2002)CrossRefADSzbMATHGoogle Scholar
  6. 6.
    M.E.J. Newman, The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003)CrossRefADSMathSciNetzbMATHGoogle Scholar
  7. 7.
    S. Boccaletti, V. Latora, Y. Moreno, M. Chávez, D.-U. Hwang, Complex networks: structure and dynamics. Phys. Rep. 424, 175–308 (2006)CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    M.E.J. Newman, M. Girvan, Finding and evaluating community structure in networks. Phys. Rev. E 69, 026113 (2004)Google Scholar
  9. 9.
    M.E.J. Newman, Fast algorithm for detecting community structure in networks. Phys. Rev. E 69, 066133 (2004)Google Scholar
  10. 10.
    M.E.J. Newman, Modularity and community structure in networks. Proc. Natl. Acad. Sci. 103, 8577–8582 (2006)CrossRefADSGoogle Scholar
  11. 11.
    A. Lancichinetti, S. Fortunato, F. Radicchi, Benchmark graphs for testing community detection algorithms. Phys. Rev. E 78, 046110 (2008)Google Scholar
  12. 12.
    D. Randall, Rapidly mixing Markov chains with applications in computer science and physics. Comput. Sci. Eng. 6, 1521–9615 (2006)Google Scholar
  13. 13.
    N. Rubido, C. Grebogi, M.S. Baptista, Structure and function in flow networks. Europhys. Lett. 101, 68001 (2013)Google Scholar
  14. 14.
    P. Erdös, A. Rényi, On random graphs I. Publ. Math. Debr. 6, 290–297 (1959)zbMATHGoogle Scholar
  15. 15.
    N. Rubido, A.C. Martí, E. Bianco-Martínez, C. Grebogi, M.S. Baptista, C. Masoller, Exact detection of direct links in networks of interacting dynamical units. New J. Phys. 16, 093010 (2014)Google Scholar
  16. 16.
    D.J. Watts, S.H. Strogatz, Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998)CrossRefADSGoogle Scholar
  17. 17.
    S. Gershgorin, Über die Abgrenzung der Eigenwerte einer Matrix. Bulletin de l’Académie des Sciences de l’URSS. Classe des sciences mathématiques et na 6, 749–754 (1931)Google Scholar
  18. 18.
    J. Cserti, Application of the lattice Green’s function for calculating the resistance of an infinite network of resistors. Am. J. Phys. 68(10), 896–906 (2000)CrossRefADSGoogle Scholar
  19. 19.
    W. Xiao, I. Gutman, Resistance distance and Laplacian spectrum. Theor. Chem. Acc. 110(4), 284–289 (2003)CrossRefGoogle Scholar
  20. 20.
    F.Y. Wu, Theory of resistor networks: the two-point resistance. J. Phys. A: Math. Gen. 37, 6653–6673 (2004)CrossRefADSzbMATHGoogle Scholar
  21. 21.
    H. Zhang, Y. Yang, Resistance distance and Kirchhoff index in circulant graphs. Int. J. Quantum Chem. 107, 330–339 (2007)CrossRefADSGoogle Scholar
  22. 22.
    H. Chen, F. Zhang, Resistance distance and the normalized Laplacian spectrum. Discret. Appl. Math. 155, 654–661 (2007)CrossRefzbMATHGoogle Scholar
  23. 23.
    A. Ghosh, S. Boyd, A. Saberi, Minimizing effective resistance of a graph. SIAM Rev. 50(1), 37–66 (2008)CrossRefADSMathSciNetzbMATHGoogle Scholar
  24. 24.
    G. Kirchhoff, in Vorlesungen über Mechanik, ed. by W. Wien (Leipzig, Germany, 1864–1928)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Universidad de la RepúblicaMontevideoUruguay

Personalised recommendations