Self-similarity in fractal and non-fractal networks

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

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)


We study the origin of scale invariance (SI) of the degree distribution in scale-free (SF) networks with a degree exponent γ under coarse graining. A varying number of vertices belonging to a community or a box in a fractal analysis is grouped into a supernode, where the box mass M follows a power-law distribution, Pm(M) ̃ M. The renormalized degree k′ of a supernode scales with its box mass M as k′ ̃ Mθ. The two exponents η and θ can be nontrivial as η ≠ γ and θ < 1. They act as relevant parameters in determining the self-similarity, i.e., the SI of the degree distribution, as follows: The self-similarity appears either when γ ≤ η or under the condition θ = (η - 1)/(γ - 1) when γ > η, irrespective of whether the original SF network is fractal or non-fractal. Thus, fractality and self-similarity are disparate notions in SF networks.

Original languageEnglish
Pages (from-to)350-356
Number of pages7
JournalJournal of the Korean Physical Society
Issue number2
Publication statusPublished - 2008 Feb


  • Coarse-graining
  • Fractality
  • Scale invariance
  • Scale-free network

ASJC Scopus subject areas

  • General Physics and Astronomy


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