# Evolution of the Complex Supply Chain Network Based on Deviation from the Power-Law Distribution
## 2 Evolutionary Characteristics and Model Analysis of Complex SCNs
### Analysis of the Deviation of the Power-Law Distribution  
An important prerequisite for previous studies on the evolution model of complex SCNs is that the total number of connected edges and the total number of nodes in the network where nodes and connected edges that grow over time are always dynamically linearly related, and the average value of the network will not be changed during the evolution degree value. However, the nonlinear accelerated growth of complex SCNs in reality is a common evolutionary phenomenon, so the following model construction and analysis are based on the scale-free accelerated growth SCN as the research object, which is more in line with the real complex supply chain evolution mechanism and form of the network. Most of the complex SCNs widely existing in reality are self-organized critical systems on the edge of chaos. An important feature of this type of network is that the degree distribution meets the power-law characteristic, that is \(p(k)\sim k^{-\gamma}\), and the value of the power index \(\gamma\) is usually less than 3. In the field of complex network research, many related models and mechanisms have been proposed to explain the important topology property of power-law distribution, such as the priority connection mechanism, adaptive model, accelerated growth model, and logarithm proposed by [PERSON] and [PERSON]. They present a growth model, random preference model, etc. [1]. However, according to a large number of empirical studies, it is found that there are few power-law distributions that completely obey factor \(k^{-\gamma}\) in the real SCN. Most of the complex SCN's degree value data only appear as a power-law distribution within the main range of the degree distribution, and the phenomenon of deviation from the power-law distribution often occurs in the range of small or large degree values. Many scholars have also discovered this kind of power-law deviation in empirical research. Some scholars proposed different models and mechanisms to explain the phenomenon of power-law distribution deviation, such as the size effect, information filtering, nonadditivity, statistics, and other mechanisms. While [PERSON] [15] and [PERSON] [16] studied the world aviation network and the business management network, respectively, they found that the cumulative degree distribution would have an exponential deviation and constructed a model describing the power-law deviation, as shown in Formula (8):  
\[p(k)=k^{-\gamma}\exp\biggl{(}\frac{k}{k_{x}}\biggr{)}, \tag{8}\]  
There is a similar exponential deviation in the citation network of scientist papers, and this phenomenon has been described since the construction of the model, as shown in Formula (9):  
\[p(x)=\frac{c_{1}}{c_{2}+(|x-x_{m}|)^{1+\alpha}}\exp(x), \tag{9}\]  
Among them, \(x\) is the number of times a paper is cited, \(x_{m}\) is the maximum probability value of \(x\), and \(c_{1}\) and \(c_{2}\) are constants. In the case where the aviation network with a two-stage power-law distribution exhibits data deviation, a mathematical equation \(p(k)=\frac{\bar{a}}{k^{\gamma_{1}}(1+bk^{\gamma_{2}})}\) can be established to represent the deviation characteristics of the power-law degree distribution of the network. In addition, there are similar deviations in the cumulative distribution of road networks in the US and Denmark. Similarly, the deviation of the power-law distribution is found in most scale-free complex SCNs, and these networks usually use truncated data sets to fit the power-law distribution model. For example, [PERSON] et al. [17] divided the degree value of the network into two parts by \(x_{\text{min}}\); the network only obeyed the power-law distribution when it was \(x\geq x_{\text{min}}\) and estimated \(x_{\text{min}}\) based on the distance test method, using the standard maximum likelihood method to estimate the value of alpha.