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

Abstract

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 ℓ needed to cover all the nodes of a cellular network was found to scale as the power-law N V ∼ ( ℓ + 1 ) - D V with a fractal dimension D V = 3.53 ± 0.26 . We implement an alternative box-counting method in terms of the minimum number N E 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 ∼ ℓ - D E with D E = 2.67 ± 0.15 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.