Advertisement

Spectral Partitioning of Random Graphs with Given Expected Degrees

  • Amin Coja-Oghlan
  • Andreas Goerdt
  • André Lanka
Part of the IFIP International Federation for Information Processing book series (IFIPAICT, volume 209)

Abstract

It is a well established fact, that—in the case of classical random graphs like (variants of) G n,p or random regular graphs-spectral methods yield efficient algorithms for clustering (e. g. coiouring or bisection) problems. The theory of large networks emerging recently provides convincing evidence that such networks, albeit looking random in some sense, cannot sensibly be described by classical random graphs. A variety of new types of random graphs have been introduced. One of these types is characterized by the fact that we have a fixed expected degree sequence, that is for each vertex its expected degree is given.

Recent theoretical work confirms that spectral methods can be successfully applied to clustering problems for such random graphs, too—provided that the expected degrees are not too small, in fact≥log6 n. In this case however the degree of each vertex is concentrated about its expectation. We show how to remove this restriction and apply spectral methods when the expected degrees are bounded below just by a suitable constant. Our results rely on the observation that techniques developed for the classical sparse G n,p random graph (that is p=c/n) can be transferred to the present situation, when we consider a suitably normalized adjacency matrix: We divide each entry of the adjacency matrix by the product of the expected degrees of the incident vertices. Given the host of spectral techniques developed for G n,p this observation should be of independent interest.

Keywords

Adjacency Matrix Random Graph Degree Distribution Cluster Problem Cluster Property 
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.

References

  1. 1.
    Aiello, W, Chung, F., Lu, L.: A random graph model for massive graphs. Proc. 33rd. SToC (2001), 171–180.Google Scholar
  2. 2.
    Alon, N. Spectral techniques in graph algorithms. Proc. LATIN (1998), LNCS 1380, Springer, 206–215.Google Scholar
  3. 3.
    Alon, N., Kahale, N.: A spectral technique for coloring random 3-colorable graphs. SIAM J. Comput. 26 (1997) 1733–1748.CrossRefMathSciNetGoogle Scholar
  4. 4.
    Boppana, R.B.: Eigenvalues and graph bisection: An average case analysis. Proc. 28th FoCS (1987), 280–285.Google Scholar
  5. 5.
    Chung, F.K.R.: Spectral Graph Theory. American Mathematical Society (1997).Google Scholar
  6. 6.
    Coja-Oghlan, A.: On the Laplacian eigenvalues of G n,p. Preprint (2005) http://www.informatik.hu-berlin.de/~coja/de/publikation.php.Google Scholar
  7. 7.
    Coja-Oghlan, A., Lanka, A.: The Spectral Gap of Random Graphs with Given Expected Degrees. Preprint (2006).Google Scholar
  8. 8.
    Chung, F.K.R., Lu, L., Vu, V.: The Spectra of Random Graphs with Given Expected Degrees. Internet Mathematics 1 (2003) 257–275.MathSciNetGoogle Scholar
  9. 9.
    Dasgupta, A., Hopcroft, J.E., McSherry, F.: Spectral Analysis of Random Graphs with Skewed Degree Distributions. Proc. 45th FOCS (2004) 602–610.Google Scholar
  10. 10.
    Feige, U., Ofek, E.: Spectral Techniques Applied to Sparse Random Graphs. Random Structures and Algorithms, 27(2) (2005), 251–275.CrossRefMathSciNetGoogle Scholar
  11. 11.
    Flaxman, A.: A spectral technique for random satisfiable 3CNF formulas. Proc. 14th SODA (2003) 357–363.Google Scholar
  12. 12.
    Friedman, J., Kahn, J., Szemeédi, E.: On the Second Eigenvalue in Random Regular Graphs. Proc. 21th STOC (1989) 587–598.Google Scholar
  13. 13.
    Fiiredi, Z., Komloś, J.: The eigenvalues of random symmetric matrices. Combinatorica 1 (1981) 233–241.MathSciNetGoogle Scholar
  14. 14.
    Giesen, J., Mitsche, D.: Reconstructing Many Partitions Using Spectral Techniques. Proe. 15th FCT (2005) 433 444.Google Scholar
  15. 15.
    Husbands, P., Simon, H., and Ding, C.: On the use of the singular value decomposition for text retrieval. In 1st SIAM Computational Information Retrieval Workshop (2000), Raleigh, NC.Google Scholar
  16. 16.
    Krivelevich, M., Sudakov, B.: The largest eigenvalue of sparse random graphs. Combinatorics, Probability and Computing 12 (2003) 61–72.CrossRefMathSciNetGoogle Scholar
  17. 17.
    Krivelevich, M., Vu, V.H.: On the concentration of eigenvalues of random symmetric matrices. Microsoft Technical Report 60 (2000).Google Scholar
  18. 18.
    Lempel, R., Moran, S. Rank-stability and rank-similarity of link-based web ranking algorithms in authority-connected graphs. Information retrieval, special issue on Advances in Mathematics/Formal methods in Information Retrieval (2004) Kluwer.Google Scholar
  19. 19.
    Meila, M., Varna D.: A comparison of spectral clustering algorithms. UW CSE Technical report 03-05-01.Google Scholar
  20. 20.
    McSherry, F.: Spectral Partitioning of Random Graphs. Proc. 42nd FoGS (2001) 529–537.Google Scholar
  21. 21.
    Mihail, M., Papadimitriou, C.H.: On the Eigenvalue Power Law. Proc. 6th RANDOM (2002) 254–262.Google Scholar
  22. 22.
    Pothen, A., Simon, H.D., Liou, K.-P.: Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Anal. Appl. 11 (1990) 430–452CrossRefMathSciNetGoogle Scholar
  23. 23.
    Spielman, D.A., Teng, S.-H.: Spectral partitioning works: planar graphs and finite element meshes. Proc. 36th FOCS (1996) 96–105.Google Scholar

Copyright information

© International Federation for Information Processing 2006

Authors and Affiliations

  • Amin Coja-Oghlan
    • 1
  • Andreas Goerdt
    • 2
  • André Lanka
    • 2
  1. 1.Institut für InformatikHumboldt Universität zu BerlinBerlinGermany
  2. 2.Fakultäit für InformatikTeehnische Universität ChemnitzChemnitzGermany

Personalised recommendations