Advertisement

Spectral Analysis of Random Networks

  • Sergei N. Dorogovtsev
  • Alexander V. Goltsev
  • José F.F. Mendes
  • Alexander N. Samukhin
Part I Network Structure
Part of the Lecture Notes in Physics book series (LNP, volume 650)

Abstract

We review a general approach that describes the spectra of eigenvalues for random graphs with a local tree-like structure. The exact equations to the spectra of networks with a local tree-like structure, are presented. The tail of the density of eigenvalues \(\rho \left( \lambda \right) \) at large \(\left| \lambda \right| \) is related to the behavior of the vertex degree distribution for large value of degree. In particular, as \( P\left( k\right) \sim k^{-\gamma }\), \(\rho \left( \lambda \right) \sim \left| \lambda \right| ^{1-2\gamma }\). Under an effective medium approximation we propose a simple approximation, calculate spectra of various graphs analytically. We also analyse the spectra of various complex networks and discuss the role of vertices of low degree. We show that spectra of locally tree-like random graphs gives a good description of the spectral properties of real-life networks like the Internet.

Keywords

Random Graph Degree Distribution Random Network Central Peak Adjacency Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1. S.N. Dorogovtsev and J.F.F. Mendes, Evolution of Networks: From Biological Nets to the Internet and WWW (Oxford, University Press, 2003).Google Scholar
  2. 2. A.-L. Barabási and R. Albert, Science 286, 509 (1999).Google Scholar
  3. 3. S.H. Strogatz, Nature 401, 268 (2001).Google Scholar
  4. 4. R. Albert and A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002).Google Scholar
  5. 5. S.N. Dorogovtsev and J.F.F. Mendes, Adv. Phys. 51, 4 (2002).Google Scholar
  6. 6. M.E.J. Newman, SIAM Review 45, 167 (2003).Google Scholar
  7. 7. D.J. Watts, Small Worlds: The Dynamics of Networks between Order and Randomness (Princeton University Press, Princeton, NJ, 1999).Google Scholar
  8. 8. Watts and Strogatz, Nature 393, 440 (1998).Google Scholar
  9. 9. D. Cvetković, M. Domb, and H. Sachs, Spectra of Graphs: Theory and Applications (Johann Ambrosius Barth, Heidelberg, 1995); D. Cvetković, P. Rowlinson, and S. Simić, Eigenspaces of graphs (Cambridge University Press, Cambridge, 1997).Google Scholar
  10. 10. F.R.K. Chung, Spectral Graph Theory (American Mathematical Society, Providence, Rhode Island, 1997).Google Scholar
  11. 11. M. Faloutsos, P. Faloutsos, C. Faloutsos, Comput. Commun. Rev., 29, 251 (1999).Google Scholar
  12. 12. G. Siganos, M. Faloutsos, P. Faloutsos, C. Faloutsos, IEEE-ACM T. Network., to appearGoogle Scholar
  13. 13. R. Monasson, Eur. Phys. J. B 12, 555 (1999).Google Scholar
  14. 14. I.J. Farkas, I. Derényi, A.-L. Barabási, and T. Vicsek, Phys. Rev. E 64, 026704 (2001); I. Farkas, I. Derenyi, H. Jeong, Z. Neda, Z.N. Oltvai, E. Ravasz, A. Schubert, A.-L. Barabási, and T. Vicsek, Physica A 314, 25 (2002).Google Scholar
  15. 15. K.-I. Goh, B. Kahng and D. Kim, Phys. Rev. E 64, 051903 (2001).Google Scholar
  16. 16. K.A. Eriksen, I. Simonsen, S. Maslov and K. Sneppen, cond-mat/0212001.Google Scholar
  17. 17. D. Vukadinović, P. Huang, and T. Erlebach, Lect. Notes Comput. Sc., 2346, 83 (2002).Google Scholar
  18. 18. O. Golinelli, cond-mat/0301437.Google Scholar
  19. 19. Th. Guhr, A. Müller-Groeling, and H.A. Weidenmüller, Phys. Rep. 299, 189 (1998); A.D. Mirlin, Phys. Rep. 326, 259 (2000).Google Scholar
  20. 20. A.J. Bray and G.J. Rodgers, Phys. Rev. B 38, 11461 (1988).Google Scholar
  21. 21. E.P. Wigner, Ann. Math. 62, 548 (1955); 65, 203 (1957); 67, 325 (1958).Google Scholar
  22. 22. G.J. Rodgers and A.J. Bray, Phys. Rev. B 37, 3557 (1988).Google Scholar
  23. 23. G. Semerjian and L.F. Cugliandolo, J. Phys. A 35, 4837 (2002).Google Scholar
  24. 24. A. Bekessy, P. Bekessy, and J. Komlos, Stud. Sci. Math. Hungar. 7, 343 (1972); E.A. Bender and E.R. Canfield, J. Combinatorial Theory A 24, 296 (1978); B. Bollobás, Eur. J. Comb. 1, 311 (1980); N.C. Wormald, J. Combinatorial Theory B 31, 156,168 (1981); M. Molloy and B. Reed, Random Structures and Algorithms 6, 161 (1995).Google Scholar
  25. 25. S.N. Dorogovtsev, J.F.F. Mendes and A.N. Samukhin, Nucl. Phys. B (2003), cond-mat/0204111.Google Scholar
  26. 26. M.E.J. Newman, Phys. Rev. Lett. 89, 208701 (2002); J. Berg and M. Lässig, Phys. Rev. Lett. 89, 228701 (2002); S.N. Dorogovtsev, J.F.F. Mendes, and A.N. Samukhin, cond-mat/0206467.Google Scholar
  27. 27. S.N. Dorogovtsev, A. V. Goltsev, J.F.F. Mendes and A.N. Samukhin, Phys. Rev. E 68, 046109 (2003).Google Scholar
  28. 28. S. Redner, A Guide to First-Passage Processes, (Cambridge University Press, Cambridge, 2001).Google Scholar
  29. 29. M. Mihail and C. Papadimitriou, Lect. Notes Comput. Sci. 254, 2483 (2002).Google Scholar
  30. 30. F. Chung, L. Lu, and V. Vu, Ann. Combinatorics, 7, 21 (2003).Google Scholar
  31. 31. R. Solomonoff and A. Rapoport, Bull. Math. Biophys. 13, 107 (1951).Google Scholar
  32. 32. P. Erdős and A. Rényi, Publications Mathematicae 6, 290 (1959); Publ. Math. Inst. Hung. Acad. Sci. 5, 17 (1960).Google Scholar
  33. 33. One should note that the position of the cut-off may depend on subtleties of the ensemble of random graphs. In particular, Z. Burda and A. Krzywicki, cond-mat/0207020, showed that the exclusion of multiple connections in a network may diminish kcut.Google Scholar
  34. 34. M. Krivelevich and B. Sudakov, Combinatorics, Probability and Computing, 12, 61 (2003).Google Scholar
  35. 35. R. Pastor-Satorras, A. Vázquez, and A. Vespignani, Phys. Rev. Lett. 87, 258701 (2001).Google Scholar
  36. 36. A. Vázquez, R. Pastor-Satorras and A. Vespignani, Phys. Rev. E 65, 066130 (2002).Google Scholar
  37. 37. S.N. Dorogovtsev, A.V. Goltsev, and J.F.F. Mendes, Phys. Rev. E 65, 066122 (2002).Google Scholar
  38. 38. S. Kirkpatrick and T.P. Eggarter, Phys. Rev. B 6, 3598 (1972).Google Scholar
  39. 39. L. He, X. Liu, and G. Strang, Stud. Appl. Math. 110, 123 (2003).Google Scholar
  40. 40. E. Ravasz, A.L. Somera, D.A. Mongru, Z.N. Oltvai, and A.-L. Barabási, Science 297, 1551 (2002).Google Scholar
  41. 41. A. Vázquez, cond-mat/0211528.Google Scholar

Authors and Affiliations

  • Sergei N. Dorogovtsev
    • 1
    • 2
  • Alexander V. Goltsev
    • 1
    • 2
  • José F.F. Mendes
    • 2
  • Alexander N. Samukhin
    • 1
    • 2
  1. 1.Departamento de Física, Universidade de Aveiro, Campus Universitário de Santiago, 3810-193 AveiroPortugal
  2. 2.A.F. Ioffe Physico-Technical Institute, 194021 St. PetersburgRussia

Personalised recommendations