# Complex Networks

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## Abstract

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.

## References

- 1.E.J. Henley, R.A. Williams,
*Graph Theory in Modern Engineering*(Academic Press, New York, 1973)zbMATHGoogle Scholar - 2.N. Biggs,
*Algebraic Graph Theory*, 2nd edn. (Cambridge University Press, New York, 1974)CrossRefzbMATHGoogle Scholar - 3.F.R.K. Chung,
*Spectral Graph Theory*(American Mathematical Society, Providence, 1997)zbMATHGoogle Scholar - 4.B. Bollobás,
*Modern Graph Theory*(Springer, New York, 1998)CrossRefzbMATHGoogle Scholar - 5.R. Albert, A.L. Barabási, Statistical mechanics of complex networks. Rev. Mod. Phys.
**74**, 47–98 (2002)CrossRefADSzbMATHGoogle Scholar - 6.M.E.J. Newman, The structure and function of complex networks. SIAM Rev.
**45**, 167–256 (2003)CrossRefADSMathSciNetzbMATHGoogle Scholar - 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.M.E.J. Newman, M. Girvan, Finding and evaluating community structure in networks. Phys. Rev. E
**69**, 026113 (2004)Google Scholar - 9.M.E.J. Newman, Fast algorithm for detecting community structure in networks. Phys. Rev. E
**69**, 066133 (2004)Google Scholar - 10.M.E.J. Newman, Modularity and community structure in networks. Proc. Natl. Acad. Sci.
**103**, 8577–8582 (2006)CrossRefADSGoogle Scholar - 11.A. Lancichinetti, S. Fortunato, F. Radicchi, Benchmark graphs for testing community detection algorithms. Phys. Rev. E
**78**, 046110 (2008)Google Scholar - 12.D. Randall, Rapidly mixing Markov chains with applications in computer science and physics. Comput. Sci. Eng.
**6**, 1521–9615 (2006)Google Scholar - 13.N. Rubido, C. Grebogi, M.S. Baptista, Structure and function in flow networks. Europhys. Lett.
**101**, 68001 (2013)Google Scholar - 14.P. Erdös, A. Rényi, On random graphs I. Publ. Math. Debr.
**6**, 290–297 (1959)zbMATHGoogle Scholar - 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.D.J. Watts, S.H. Strogatz, Collective dynamics of ‘small-world’ networks. Nature
**393**, 440–442 (1998)CrossRefADSGoogle Scholar - 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.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.W. Xiao, I. Gutman, Resistance distance and Laplacian spectrum. Theor. Chem. Acc.
**110**(4), 284–289 (2003)CrossRefGoogle Scholar - 20.F.Y. Wu, Theory of resistor networks: the two-point resistance. J. Phys. A: Math. Gen.
**37**, 6653–6673 (2004)CrossRefADSzbMATHGoogle Scholar - 21.H. Zhang, Y. Yang, Resistance distance and Kirchhoff index in circulant graphs. Int. J. Quantum Chem.
**107**, 330–339 (2007)CrossRefADSGoogle Scholar - 22.H. Chen, F. Zhang, Resistance distance and the normalized Laplacian spectrum. Discret. Appl. Math.
**155**, 654–661 (2007)CrossRefzbMATHGoogle Scholar - 23.A. Ghosh, S. Boyd, A. Saberi, Minimizing effective resistance of a graph. SIAM Rev.
**50**(1), 37–66 (2008)CrossRefADSMathSciNetzbMATHGoogle Scholar - 24.G. Kirchhoff, in
*Vorlesungen über Mechanik*, ed. by W. Wien (Leipzig, Germany, 1864–1928)Google Scholar

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