  Song, Havlin and Makse (2005) have recently used a version of the
box-counting method, called the node-covering method, to quantify the
self-similar properties of 43 cellular networks: the minimal number $N_V$ of
boxes of size $\ell$ needed to cover all the nodes of a cellular network was
found to scale as the power law $N_V \sim (\ell+1)^{-D_V}$ with a fractal
dimension $D_V=3.53\pm0.26$. We propose a new box-counting method based on
edge-covering, which outperforms the node-covering approach when applied to
strictly self-similar model networks, such as the Sierpinski network. The
minimal number $N_E$ of boxes of size $\ell$ in the edge-covering method is
obtained with the simulated annealing algorithm. We take into account the
possible discrete scale symmetry of networks (artifactual and/or real), which
is visualized in terms of log-periodic oscillations in the dependence of the
logarithm of $N_E$ as a function of the logarithm of $\ell$. In this way, we
are able to remove the bias of the estimator of the fractal dimension, existing
for finite networks. With this new methodology, we find that $N_E$ scales with
respect to $\ell$ as a power law $N_E \sim \ell^{-D_E}$ with $D_E=2.67\pm0.15$
for the 43 cellular networks previously analyzed by Song, Havlin and Makse
(2005). Bootstrap tests suggest that the analyzed cellular networks may have a
significant log-periodicity qualifying a discrete hierarchy with a scaling
ratio close to 2. In sum, we propose that our method of edge-covering with
simulated annealing and log-periodic sampling minimizes the significant bias in
the determination of fractal dimensions in log-log regressions.
