### Abstract

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, P_{m}(M) ̃ M^{-η}. The renormalized degree k′ of a supernode scales with its box mass M as k′ ̃ M^{θ}. 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 language | English |
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Pages (from-to) | 350-356 |

Number of pages | 7 |

Journal | Journal of the Korean Physical Society |

Volume | 52 |

Issue number | 2 |

Publication status | Published - 2008 Feb 1 |

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### Keywords

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

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Journal of the Korean Physical Society*,

*52*(2), 350-356.