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