The topology of scale-free networks with an S-shaped nonlinear growth characteristic

Calculations were conducted concerning the degree distribution functions of four models with an S-shaped growth characteristic and the preferential attachment rule, based on the mean field theory. Of these models, Model 1 displays the simplest sigmoid function, Model 2 and Model 3 are two extended models, and Model 4 depicts the law of population function. The results show that the graphs of the four degree distribution functions with their different gained control parameters are relatively similar to one another, thus, displaying a power-law form. In addition, a separated method was defined to calculate the degree distribution of single-peak, real-time networks with a symmetric or asymmetric characteristic. Such findings, to a large extent, will enrich the complex theory and could be conducive to understanding the evolutionary dynamics of S-shaped functions as well as other exponential functions.