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The Spectral Gap of Random Graphs with Given Expected Degrees

  • Amin Coja-Oghlan
  • André Lanka
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4051)

Abstract

We investigate the Laplacian eigenvalues of a random graph G(n,d) with a given expected degree distribution d. The main result is that w.h.p. G(n,d) has a large subgraph core(G(n,d)) such that the spectral gap of the normalized Laplacian of core(G(n,d)) is \(\geq1-c_0{\bar d}_{\min}^{-1/2}\) with high probability; here c 0>0 is a constant, and \({\bar d}_{\min}\) signifies the minimum expected degree. This result is of interest in order to extend known spectral heuristics for random regular graphs to graphs with irregular degree distributions, e.g., power laws. The present paper complements the work of Chung, Lu, and Vu [Internet Mathematics 1, 2003].

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References

  1. 1.
    Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Reviews of modern physics 74, 47–97 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Alon, N., Kahale, N.: A spectral technique for coloring random 3-colorable graphs. SIAM J. Comput. 26, 1733–1748 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alon, N., Krivelevich, M., Sudakov, B.: Finding a large hidden clique in a random graph. Random Structures and Algorithms 13, 457–466 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bollobas, B., Riordan, O.: Mathematical results on scale-free random graphs. In: Bornholdt, S., Schuster, H.G. (eds.) Handbook of graphs and networks: from the genome to the Internet, pp. 1–34. Wiley, Chichester (2003)Google Scholar
  5. 5.
    Chung, F.: Spectral Graph Theory. American Mathematical Society (1997)Google Scholar
  6. 6.
    Chung, F., Lu, L., Vu, V.: The spectra of random graphs with given expected degrees. Internet Mathematics 1, 257–275 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Coja-Oghlan, A.: On the Laplacian eigenvalues of Preprint (2005)Google Scholar
  8. 8.
    Dasgupta, A., Hopcroft, J.E., McSherry, F.: Spectral Partitioning of Random Graphs. In: Proc. 45th FOCS, pp. 529–537 (2004)Google Scholar
  9. 9.
    Faloutsos, M., Faloutsos, P., Faloutsos, C.: On powerlaw relationships of the internet topology. In: Proc.of ACM-SIGCOMM, pp. 251–262 (1999)Google Scholar
  10. 10.
    Feige, U., Ofek, E.: Spectral techniques applied to sparse random graphs. Random Structures and Algorithms 27, 251–275 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Friedman, J., Kahn, J., Szemeredi, E.: On the second eigenvalue in random regular graphs. In: Proc.21st STOC, pp. 587–598 (1989)Google Scholar
  12. 12.
    Füredi, Z., Komloś, J.: The eigenvalues of random symmetric matrices. Combinatorica 1, 233–241 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Krivelevich, M., Vu, V.H.: Approximating the independence number and the chromatic number in expected polynomial time. J.of Combinatorial Optimization 6, 143–155 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    McSherry, F.: Spectral partitioning of random graphs. In: Proc.42nd FOCS, pp. 529–537 (2001)Google Scholar
  15. 15.
    Mihail, M., Papadimitriou, C.H.: On the eigenvalue power law. In: Rolim, J.D.P., Vadhan, S.P. (eds.) RANDOM 2002. LNCS, vol. 2483, pp. 254–262. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  16. 16.
    Pothen, A., Simon, H.D., Liou, K.-P.: Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Anal.Appl. 11, 430–452 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Schloegel, K., Karypis, G., Kumar, V.: Graph partitioning for high performance scientific simulations. In: Dongarra, J., Foster, I., Fox, G., Kennedy, K., White, A. (eds.) CRPC parallel computation handbook, Morgan Kaufmann, San Francisco (2000)Google Scholar
  18. 18.
    Wigner, E.P.: On the distribution of the roots of certain symmetric matrices. Annals of Mathematics 67, 325–327 (1958)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Amin Coja-Oghlan
    • 1
  • André Lanka
    • 2
  1. 1.Institut für InformatikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Fakultät für InformatikTechnische Universität ChemnitzChemnitzGermany

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