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

Song et al. [Self-similarity of complex networks, Nature 433 (2005) 392-395] 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 C needed to cover all the nodes of a cellular network was found to scale as the power-law N-V similar to(l +1)(-Dv) with a fractal dimension D-V = 3.53 +/- 10.26. We implement an alternative box-counting method in terms of the minimum number NE of edge-covering boxes which is well-suited to cellular networks, where the search over different covering sets is performed with the simulated annealing algorithm. The method also takes into account a possible discrete scale symmetry to optimize the sampling rate and minimize possible biases in the estimation of the fractal dimension. With this methodology, we find that N-E scales with respect to l as a power-law N-E similar to l(DE) with DE = 2.67 +/- 10.15 for the 43 cellular networks previously analyzed by Song et al. [Self-similarity of complex networks, Nature 433 (2005) 392-395]. 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. (c) 2006 Elsevier B.V. All rights reserved.