Self-similarity in fractal and non-fractal networks

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.