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Wave Localization on Complex Networks

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Towards an Information Theory of Complex Networks

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Abstract

In this chapter we consider the role played by Anderson localization in transport through complex networks, for example, an optical network. The network is described by a tight binding Hamiltonian, which may be used to determine the properties of the Anderson transition according to the statistical properties of its eigenvalues. The Anderson transition properties of different complex networks will be studied, emphasizing the role played by clustering on the localization of waves. We shall show that new complex topologies lead to novel physics, specifically clustering may lead to localization.

MSC2000 Primary 05C80; Secondary 90B18, 90B15, 05C50, 05C82.

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Notes

  1. 1.

    The N dependence is well established [35] but the C 0 dependence seems to be unexplored. Our data for N up to 105 suggest \(\ln a \propto {\left ({C}_{0} - {C}_{0,c}\right )}^{-{\nu }_{c}}\). Since C 0, q < C 0, c, a and thus L depend weakly on C 0 at the quantum transition.

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Author information

Authors and Affiliations

  1. Minerva Center and Department of Physics, Bar-Ilan University, Ramat-Gan, 52900, Israel

    Richard Berkovits

  2. Martin-Luther-Universität Halle-Wittenberg, 06099, Halle, Germany

    Lukas Jahnke & Jan W. Kantelhardt

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  1. Richard Berkovits
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  2. Lukas Jahnke
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  3. Jan W. Kantelhardt
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Correspondence to Richard Berkovits .

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Editors and Affiliations

  1. Medizinische Informatik und Technik, Institute for Bioinformatics and Transla, UMIT-Private Universität für Gesundheits, Eduard Wallnöfer-Zentrum 1, Hall in Tirol, A-6060, Austria

    Matthias Dehmer

  2. Queen's University Belfast, School of Medicine, Dentistry, and Cell, Center for Cancer Research & Cell Biolog, Lisburn Road 97, Belfast, BT97BL, United Kingdom

    Frank Emmert-Streib

  3. Goethe-University Frankfurt am Main, Department of Philosophy and Historical, Center for Computing in the Humanities, Senckenberganlage 31, Frankfurt, 60325, Germany

    Alexander Mehler

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Berkovits, R., Jahnke, L., Kantelhardt, J.W. (2011). Wave Localization on Complex Networks. In: Dehmer, M., Emmert-Streib, F., Mehler, A. (eds) Towards an Information Theory of Complex Networks. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4904-3_4

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