Wave Localization on Complex Networks

  • Richard Berkovits
  • Lukas Jahnke
  • Jan W. Kantelhardt


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.


Anderson localization Complex and random graphs Information theory 


  1. 1.
    Anderson, P.W.: Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958)CrossRefGoogle Scholar
  2. 2.
    Kramer, B., MacKinnon, A.: Localization – theory and experiment. Rep. Prog. Phys. 56, 1496–1564 (1993)CrossRefGoogle Scholar
  3. 3.
    Wiersma, D.S., Bartolini, P., Lagendijk, A., Righini, R.: Localization of light in disordered medium. Nature 390, 671–673 (1997)CrossRefGoogle Scholar
  4. 4.
    Störzer, M., Gross, P., Aegerter, C.M., Maret, G.: Observation of the critical regime near Anderson localization of light. Phys. Rev. Lett. 96, 063904 (2006)CrossRefGoogle Scholar
  5. 5.
    Schwartz, T., Bartal, G., Fishman, S., Segev, M.: Transport and Anderson localization in disordered two-dimensional photonic lattices. Nature 446, 52–55 (2007)CrossRefGoogle Scholar
  6. 6.
    Lahini, Y., Avidan, A., Pozzi, F., Sorel, M., Morandotti, R., Christodoulides, D.N., Silberberg, Y.: Anderson localization and nonlinearity in one-dimensional disordered photonic lattices. Phys. Rev. Lett. 100, 013906 (2008)CrossRefGoogle Scholar
  7. 7.
    Foret, M., Courtens, E., Vacher, R., Suck, J.B.: Scattering investigation of acoustic localization in fused silica. Phys. Rev. Lett. 77, 3831–3834 (1996)CrossRefGoogle Scholar
  8. 8.
    Kantelhardt, J.W., Bunde, A., Schweitzer, L.: Extended fractons and localized phonons on percolation clusters. Phys. Rev. Lett. 81, 4907–4910 (1998)CrossRefGoogle Scholar
  9. 9.
    Billy, J., Josse, V., Zuo, Z.C., Bernard, A., Hambrecht, B., Lugan, P., Clement, D., Sanchez-Palencia, L., Bouyer, P., Aspect, A.: Direct obeservation of Anderson localization of matter waves in a controlled disorder. Nature 453, 891 (2008)CrossRefGoogle Scholar
  10. 10.
    Roati, G., D’Errico, C., Fallani, L., Fattori, M., Fort, C., Zaccanti, M., Modugno, G., Modugno, M., Inguscio, M.: Anderson localization of a non-interacting bose-einstein condensate. Nature 453, 895–898 (2008)CrossRefGoogle Scholar
  11. 11.
    Abrahams, E., Anderson, P.W., Licciardello, D.C., Ramakrishnan, T.V.: Scaling theory of localization – absence of quantum diffusion in 2 dimensions. Phys. Rev. Lett. 42, 673–676 (1979)CrossRefGoogle Scholar
  12. 12.
    Castellani, C., DiCastro, C., Peliti, L.: On the upper critical dimension in Anderson localization. J. Phys. A 19, 1099–1103 (1986)CrossRefGoogle Scholar
  13. 13.
    Kunz, H., Souillard, B.: On the upper critical dimension and the critical exponents of the localization transition. J. Phys. Lett. 44, L503–L506 (1983)CrossRefGoogle Scholar
  14. 14.
    Straley, J.P.: Conductivity near the localization threshold in the high-dimensionality limit. Phys. Rev. B 28, 5393 (1983)CrossRefGoogle Scholar
  15. 15.
    Lukes, T.: Critical dimensionality in the Anderson-Mott transition. J. Phys. C 12, L797 (1979)CrossRefGoogle Scholar
  16. 16.
    Efetov, K.B.: Anderson transition on a bethe lattice (the symplectic and orthogonal ensembles). Zh. Eksp. Teor. Fiz 93, 1125–1139 (1987) [Sov. Phys. JETP, 61, 606 (1985)]Google Scholar
  17. 17.
    Zhu, C.P., Xiong, S.-J.: Localization-delocalization transition of electron states in a disordered quantum small-world network. Phys. Rev. B 62, 14780 (2000)CrossRefGoogle Scholar
  18. 18.
    Giraud, O., Georgeot, B., Shepelyansky, D.L.: Quantum computing of delocalization in small-world networks. Phys. Rev. E 72, 036203 (2005)CrossRefGoogle Scholar
  19. 19.
    Gong, L., Tong, P.: von Neumann entropy and localization-delocalization transition of electron states in quantum small-world networks. Phys. Rev. E 74, 056103 (2006)CrossRefGoogle Scholar
  20. 20.
    Sade, M., Berkovits, R.: Localization transition on a cayley tree via spectral statistics. Phys. Rev. B 68, 193102 (2003)CrossRefGoogle Scholar
  21. 21.
    Sade, M., Kalisky, T., Havlin, S., Berkovits, R.: Localization transition on complex networks via spectral statistics. Phys. Rev. E 72, 066123 (2005)CrossRefGoogle Scholar
  22. 22.
    Shapiro, B.: Renormalization-group transformation for the Anderson transition. Phys. Rev. Lett. 48, 823–825 (1982)CrossRefGoogle Scholar
  23. 23.
    Anderson, P.W., Thouless, D.J., Abrahams, E., Fisher, D.S.: New Method for a scaling theory of localization. Phys. Rev. B 22, 3519–3526 (1980)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Edrei, I., Kaveh, M., Shapiro, B.: Probability-distribution functions for transmission of waves through random-media – a new numerical-method. Phys. Rev. Lett. 62, 2120–2123 (1989)CrossRefGoogle Scholar
  25. 25.
    Carmi, S., Havlin, S., Kirkpatrick, S., Shavitt, Y., Shir, E.: A model of internet topology using k-shell decomposition. PNAS 104, 11150–11154 (2007)CrossRefGoogle Scholar
  26. 26.
    Vázquez, A., Pastor-Satorras, R., Vespignani, A.: Large-scale topological and dynamical properties of the internet. Phys. Rev. E 65, 066130 (2002)CrossRefGoogle Scholar
  27. 27.
    Shklovskii, B.I., Shapiro, B., Sears, B.R., Lambrianides, P., Shore, H.B.: Statistics of spectra of disordered-systems near the metal-insulator-transition. Phys. Rev. B 47, 11487–11490 (1993)CrossRefGoogle Scholar
  28. 28.
    Hofstetter, E., Schreiber, M.: Relation between energy-level statistics and phase transition and its application to the Anderson model. Phys. Rev. E 49, 14726 (1994)Google Scholar
  29. 29.
    For a recent review see: Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)Google Scholar
  30. 30.
    Erdös, P., Rényi, A.: On random graphs. Publ. Math. Debrecen 6, 290–297 (1959)MATHGoogle Scholar
  31. 31.
    Albert, R., Jeong, H., Barabási, A.L.: Error and attack tolerance of complex networks. Nature 406, 378–382 (2000)CrossRefGoogle Scholar
  32. 32.
    Kalisky, T. , Cohen, R. , ben Avraham, D., Havlin, S.: Tomography and stability of complex networks. In: Ben-Naim, E., Frauenfelder, H., Toroczkai, Z. (eds.) Lecture Notes in Physics: Proceedings of the 23rd LANL-CNLS Conference, “Complex Networks”, Santa-Fe, 2003. Springer, Berlin (2004)Google Scholar
  33. 33.
    Cohen, R., Erez, K., ben Avraham, D., Havlin, S.: Resilience of the internet to random breakdowns. Phys. Rev. Lett. 85, 4626–4628 (2000)Google Scholar
  34. 34.
    Cohen, R., Erez, K., ben Avraham, D., Havlin, S.: Resilience of the internet to random breakdowns. Phys. Rev. Lett. 86, 3682 (2001)Google Scholar
  35. 35.
    Cohen, R., Havlin, S.: Scale-free networks are ultrasmall. Phys. Rev. Lett. 90, 058701 (2003)CrossRefGoogle Scholar
  36. 36.
    Bollobas, B., Riordan, O.: Mathematical results on scale-free random graphs. In: Bornholdt, S., Schuster, H.G. (eds.) Handboook of Graphs and Networks. Wiley-VCH, Berlin (2002)Google Scholar
  37. 37.
    Molloy, M., Reed, B.: The size of the giant component of a random graph with a given degree sequence. Combinator. Probab. Comput. 7, 295–305 (1998)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Berkovits, R., Avishai, Y.: Spectral statistics near the quantum percolation threshold. Phys. Rev. B 53, R16125–R16128 (1996)CrossRefGoogle Scholar
  39. 39.
    Kopp, A., Jia, X., Chakravarty, S.: Replacing energy by von Neumann entropy in quantum phase transitions. Ann. Phys. 322, 1466–1476 (2007)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Lorenz, C.D., Ziff, R.M.: Precise Determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and bcc lattices. Phys. Rev. E 57, 230–236 (1998)CrossRefGoogle Scholar
  41. 41.
    Schreiber, M., Grussbach, H.: Dimensionality dependence of the metal-insulator transition in the Anderson model of localization. Phys. Rev. Lett. 76, 1687–1690 (1996)CrossRefGoogle Scholar
  42. 42.
    Jahnke, L., Kantelhardt, J.W., Berkovits, R., Havlin, S.: Wave localization in complex networks with high clustering. Phys. Rev. Lett. 101, 175702 (2008)CrossRefGoogle Scholar
  43. 43.
    Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of Networks – From Biological Nets to the Internet and WWW. Oxford University Press, Oxford (2003)MATHGoogle Scholar
  44. 44.
    Pastor-Satorras, R., Vespignani, A.: Evolution and Structure of the Internet: A Statistical Physics Approach. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  45. 45.
    Newman, M.E.J.: Assortative mixing in networks. Phys. Rev. Lett. 89, 208701 (2002)CrossRefGoogle Scholar
  46. 46.
    Watts, D.J, Strogatz, S.H.: Collective dynamics of ’small-world’ networks. Nature 393, 440–442 (1998)Google Scholar
  47. 47.
    Serrano, M.A., Boguñá, M.: Percolation and epidemic thresholds in clustered networks. Phys. Rev. Lett. 97, 088701 (2006)CrossRefGoogle Scholar
  48. 48.
    Serrano, M.A., Boguñá, M.: Clustering in complex networks. I. General formalism. Phys. Rev. E 74, 056114 (2006)CrossRefGoogle Scholar
  49. 49.
    Serrano, M.A., Boguñá, M.: Clustering in complex networks. II. Percolation properties. Phys. Rev. E 74, 056115 (2006)Google Scholar
  50. 50.
    Serrano, M.A., Boguñá, M.: Tuning clustering in random networks with arbitrary degree distributions. Phys. Rev. E 72, 036133 (2005)CrossRefGoogle Scholar
  51. 51.
    Volz, E.: Random networks with tunable degree distribution and clustering. Phys. Rev. E 70, 056115 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Richard Berkovits
    • 1
  • Lukas Jahnke
    • 2
  • Jan W. Kantelhardt
    • 2
  1. 1.Minerva Center and Department of PhysicsBar-Ilan UniversityRamat-GanIsrael
  2. 2.Martin-Luther-Universität Halle-WittenbergHalleGermany

Personalised recommendations