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 gamma 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) similar to M-eta. The renormalized degree k' of a supernode scales with its box mass M as k' similar to M-theta. The two exponents eta and theta can be nontrivial as n not equal gamma and theta < 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 gamma <= eta or under the condition theta = (eta - 1)/(gamma - 1) when gamma > eta, irrespective of whether the original SF network is fractal or non-fractal. Thus, fractality and self-similarity are disparate notions in SF networks.