Nonextensive aspects of small-world networks

Nonextensive aspects of the degree distribution in Watts-Strogatz (WS) small-world networks, $P_{SW}(k)$, 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 $P_{SW}(k)$ 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.