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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 of
boxes of size needed to cover all the nodes of a cellular network was
found to scale as the power law with a fractal
dimension . 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 of boxes of size 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 as a function of the logarithm of . 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 scales with
respect to as a power law with
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.Comment: 19 elsart pages including 9 eps figure
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