Nonextensive aspects of small-world networks

Hideo Hasegawa

doi:10.1016/j.physa.2005.10.004

paper

Nonextensive aspects of small-world networks

arXiv:cond-mat/0506301 · doi:10.1016/j.physa.2005.10.004

Abstract

Nonextensive aspects of the degree distribution in Watts-Strogatz (WS) small-world networks, $P_{S W} (k)$ , have been discussed in terms of a generalized Gaussian (referred to as {\it $Q$ -Gaussian}) which is derived by the three approaches: the maximum-entropy method (MEM), stochastic differential equation (SDE), and hidden-variable distribution (HVD). In MEM, the degree distribution $P_{Q} (k)$ in complex networks has been obtained from $Q$ -Gaussian by maximizing the nonextensive information entropy with constraints on averages of $k$ and $k^{2}$ in addition to the normalization condition. In SDE, $Q$ -Gaussian is derived from Langevin equations subject to additive and multiplicative noises. In HVD, $Q$ -Gaussian is made by a superposition of Gaussians for random networks with fluctuating variances, in analogy to superstatistics. Interestingly, {\it a single} $P_{Q} (k)$ may describe, with an accuracy of $\mid P_{SW}(k)-P_Q(k)\mid \siml 10^{-2} $, main parts of degree distributions of SW networks, within which about 96-99 percents of all $k$ states are included. It has been demonstrated that the overall behavior of $P_{S W} (k)$ including its tails may be well accounted for if the $k$ -dependence is incorporated into the entropic index in MEM, which is realized in microscopic Langevin equations with generalized multiplicative noises.

22 pages, 11 figures, accepted in Physca A with some augmentations

Nonextensive aspects of small-world networks

Abstract

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