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, Pm(M) ∼ M −� . The renormalized degree kof a supernode scales with its box mass M as k ' ∼ M � . The two exponents η and θ can be nontrivial as η 6 γ and θ η, irrespective of whether the original SF network is fractal or non-fractal. Thus, fractality and self-similarity are disparate notions in SF networks.