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Register a new userExploring self-similarity of complex cellular networks: The edge-covering method with simulated annealing and log-periodic sampling
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Wei-Xing Zhou
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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 $D_V=3.53pm0.26$. 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 $D_E=2.67pm0.15$ 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.
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