Exploring self-similarity of complex cellular networks: the edge-covering method with simulated annealing and log-periodic sampling

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 NV of boxes of size l needed to cover all the nodes of a cellular network was found to scale as the power law NV ∼ (l+1) −DV with a fractal dimension DV = 3.53±0.26. 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 NE of boxes of size l 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 logperiodic oscillations in the dependence of the logarithm of NE as a function of the logarithm of l. 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 NE scales with respect to l as a power law NE ∼ l −DE with DE = 2.67± 0.15 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.