# Evolution of the Complex Supply Chain Network Based on Deviation from the Power-Law Distribution
## 4 Simulation of SCN Evolution
### Simulation Analysis of Network Evolution Based on Deviation of Two-Stage Power Law Distribution  
In real life, most complex SCNs will exhibit different dual-segment power exponential distributions over time, mainly due to the inherent driving force of the growth of complex SCNs in various fields and different external environments, which in turn leads to significant differences in the evolution parameters of the network, mainly reflected in the probability parameter \(p_{1}\), \(p_{2}\), \(q\), \(q\)/ and the power index \(\gamma_{1}\), \(\gamma_{2}\) and other parameter values. This type of SCN is generally composed of upstream raw material suppliers, component suppliers and other auxiliary product production node companies. Midstream companies are manufacturing products and processing node companies, while downstream companies mainly include transportation node companies and product demand nodal enterprises and individuals. Upstream, midstream, and downstream nodal enterprises in the network constitute an interconnected and codeveloped topology system, and the system is always dynamic over time. In this section, based on the previous model construction analysis, combined with numerical experiments and model simulation methods, and then based on the contrast analysis of the two-stage power law distribution deviation phenomenon, the network evolution and the distribution degree and characteristics of node degree values under different second-stage power index conditions are analyzed. In the process of setting up numerical experiments, the scale of the SCN is set to a medium-sized network, the number of nodes is \(N{=}\ 1\times 10^{3}\) and other parameters of the evolution model are: \(t{=}\ 5\times 10^{3}\), \(m=2\), \(p_{1}=1\), \(n=3\), \(p_{2}=1\), \(c=2\times 10^{-3}\). In addition, different types of line shapes are used to characterize the simulation results of models with different power indexes. The second-stage power index corresponding to the solid line is \(\gamma_{2}{=}\ 1.5\), the power index corresponding to the dotted line is \(\gamma_{2}{=}\ 2.5\), and the power index corresponding to the dotted line is \(\gamma_{2}{=}\ 3\). In addition, the parameter values corresponding to the three types of legends are different: the parameter corresponding to the dot is \(q{=}\ 1\), \(q{=}\ 0\), the parameter corresponding to the triangle is \(q{=}\ 0.5\), \(q{=}\ 0.5\), and the parameter corresponding to the square is \(q{=}\ 0\), \(q{=}\ 0\). In the case where the three power exponents are different, the numerical experiment and model simulation process are performed separately with the help of Python software, and the two calculation results are integrated into one graph. The final result is shown in Figure 6. It can be found from the figure that the two results can be fitted well and have significant two-stage power law distribution characteristics. At this time, the average degree values of the network are 1.45, 1.76, and 1.99, respectively, and the value of the curve turning point \(k_{c}\) of the two-segment power law distribution is 11, 13.8, and 18.6, respectively. Among them, P(k) is the probability density function of the SCN node degree value k. It can be seen from the above figure that when the power exponent of the second segment of the distribution curve is \(\gamma_{2}{=}\ 1.5\), the second segment of the network node degree value distribution curve has the same slope as the first segment. A straight line is a single power law distribution. As the power exponent \(\gamma_{2}\) of the second distribution curve continues to increase, its slope also gradually increases, which indicates that the number of nodes with larger degrees in the SCN changes faster, i.e., the number of nodes with larger degrees will accelerate. At the same time, the turning point \(k_{c}\) of the corresponding two-segment power law distribution curve of the SCN also gradually becomes larger, which is reflected in the figure as gradually moving to the right.