Nonextensive aspects of small - world networks 1

We have discussed the nonextensive aspects of degree distribution P (k) in Watts-Strogatz (WS) small-world networks by using the three approaches: (a) the maximum-entropy method, (b) hidden-variable distribution and (c) stochastic differential equation. In the method (a), P (k) in complex networks has been obtained by maximizing the nonexten-sive information entropy with the constraint of < k 2 >= ρ in addition to < k >= µ, where < · > denotes the average over P (k). The distribution P (k) is expressed by a generalized Gaussian (referred to as Q-Gaussian) which has a maximum at k = µ with a width proportional to √ ρ − µ 2. In the method (b), Q-Gaussian is shown to be made by a superposition of Gaussians with fluctuating variances: WS small-world networks are constructed from random networks, in analogy to superstatistics. In the method (c), Q-Gaussian is derived from Langevin equation subject to additive and multiplicative noises. It has been demonstrated that our Q-Gaussian well describes degree distributions of WS small-world networks obtained by simulations and analytical method.