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)


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.


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.


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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

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