Scale-free random graphs and Potts model

D. S. Lee, K. I. Goh, B. Kahng, D. Kim

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)


We introduce a simple algorithm that constructs scale-free random graphs efficiently: each vertex i has a prescribed weight Pii-μ (0 < μ < 1) and an edge can connect vertices i and j with rate PiPj. Corresponding equilibrium ensemble is identified and the problem is solved by the q → 1 limit of the q-state Potts model with inhomogeneous interactions for all pairs of spins. The number of loops as well as the giant cluster size and the mean cluster size are obtained in the thermodynamic limit as a function of the edge density. Various critical exponents associated with the percolation transition are also obtained together with finite-size scaling forms. The process of forming the giant cluster is qualitatively different between the cases of λ > 3 and 2 < λ < 3, where λ = 1 + μ-1 is the degree distribution exponent. While for the former, the giant cluster forms abruptly at the percolation transition, for the latter, however, the formation of the giant cluster is gradual and the mean cluster size for finite N shows double peaks.

Original languageEnglish
Pages (from-to)1149-1159
Number of pages11
JournalPramana - Journal of Physics
Issue number6 SPEC. ISS.
Publication statusPublished - 2005 Jun
Externally publishedYes


  • Percolation transition
  • Potts model
  • Scale-free random graph

ASJC Scopus subject areas

  • Physics and Astronomy(all)


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