Song, Havlin and Makse (2005) have recently used a version of thebox-counting method, called the node-covering method, to quantify theself-similar properties of 43 cellular networks: the minimal number $N_V$ ofboxes of size $\ell$ needed to cover all the nodes of a cellular network wasfound to scale as the power law $N_V \sim (\ell+1)^{-D_V}$ with a fractaldimension $D_V=3.53\pm0.26$. We propose a new box-counting method based onedge-covering, which outperforms the node-covering approach when applied tostrictly self-similar model networks, such as the Sierpinski network. Theminimal number $N_E$ of boxes of size $\ell$ in the edge-covering method isobtained with the simulated annealing algorithm. We take into account thepossible discrete scale symmetry of networks (artifactual and/or real), whichis visualized in terms of log-periodic oscillations in the dependence of thelogarithm of $N_E$ as a function of the logarithm of $\ell$. In this way, weare able to remove the bias of the estimator of the fractal dimension, existingfor finite networks. With this new methodology, we find that $N_E$ scales withrespect 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 asignificant log-periodicity qualifying a discrete hierarchy with a scalingratio close to 2. In sum, we propose that our method of edge-covering withsimulated annealing and log-periodic sampling minimizes the significant bias inthe determination of fractal dimensions in log-log regressions.
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