Nonextensive aspects of small-world networks

Nonextensive aspects of the degree distribution in Watts-Strogatz (WS) small-world networks, PSW(k), have been discussed in terms of a generalized Gaussian (referred to as 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 PQ(k) in complex networks has been obtained from Q-Gaussian by maximizing the nonextensive information entropy with constraints on averages of k and k2 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, a single PQ(k) may describe, with an accuracy of |PSW(k)-PQ(k)|≲10-2, main parts of degree distributions of SW networks, within which about 96-99% of all k states are included. It has been demonstrated that the overall behavior of PSW(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.