The Asymptotic Mandelbrot Law of Some Evolution Networks

Complex networks are now the joint focus of many branches of research. Particularly, the scale-free property of some networks attracts continuous interests, due to their importance and pervasiveness. In short, this property means that the degree distribution of a network obeys a power law P (k) ∝ k , where k is the degree and P (k) is the corresponding probability density, and the scaling exponent γ is a constant. A pioneering model that generates power-law degree distribution was presented by Barabási and Albert (BA). In recent studies, it was found that in some complex networks, e.g. transportation networks and social collaboration networks, the degree distribution follows the so-called “shifted power law” P (k) ∝ (k + c) , where the shifting coefficient c is another constant. This property is also called “Mandelbrot law”. To understand the origins of such Mandelbrot law, Ren, Yang and Wang proposed a interesting growing network that is generated with linear preferential attachment. In such networks, there exits a recursive dependence relationship between every two consecutive degrees