Self-similarity in Fractal and Non-fractal Networks

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, $P_m(M)\sim M^{-η}$. The renormalized degree $k^{\prime}$ of a supernode scales with its box mass $M$ as $k^{\prime} \sim M^θ$. The two exponents $η$ and $θ$ can be nontrivial as $η\ne γ$ 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 $γ\le η$ 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.