# Evolution of the Complex Supply Chain Network Based on Deviation from the Power-Law Distribution
## 3 Evolution Model of a Complex SCN Based on the Deviation of the Power Law Distribution
### Analysis of SCN Evolution Model Parameters  
Based on the above analysis of the complex SCN evolution model, it can be seen that the evolution of the network on the time scale is subject to multiple model parameters. Next, we further analyze the influence of the parameters on the evolution of the SCN by controlling some key parameters of the model, such as the probability of adding a new node \(p_{1}\), the probability of adding a new connected edge \(p_{2}\), and the number of connected edges \(n\). * Probability of joining new edges \(p_{2}\)  
To further analyze the dynamic evolution of the complex SCN when there is no increase in the connected edges in the network, this section sets the probability of joining the new connected edge \(p_{2}\) to 0. Based on continuity theory, Formula (13) can be transformed as follows:  
\[\frac{\partial k_{i}}{\partial t}=\frac{mp_{1}k_{i}}{\frac{\sum k_{j}}{j}}+ \frac{nqk_{i}}{\sum k_{j}}-\frac{nq}{p_{1}t}, \tag{20}\]  
Among them, the total connection value \(\sum\limits_{j}k_{j}=(2 mp_{1}-nq+nq^{\prime})t\) in the network. The first item on the right side of the medium number in the above formula represents the process of adding nodes during the evolution of the network, and the last two items are the process of reconnecting the edges in the network. When the time step is \(t_{i}\), combined with the initial condition \(k_{i}(t_{i})=m\), the expression of the solution to Formula (20) is:  
\[k_{i}(t)=(\frac{t}{t_{i}})^{\alpha}(m-\frac{qnf}{p_{1}(mp_{1}+nq)})+\frac{qnf }{p_{1}(mp_{1}+nq^{\prime})}, \tag{21}\]  
The above formula is the scale expression of the SCN at time \(t_{i}\). Therefore, the probability density expression \(P(k,t)\) of the network node degree distribution is:  
\[P(k)=\frac{\partial p(k_{i}(t)<k)}{\partial k}=\frac{f}{mp_{1}+nq^{\prime}}(m -\frac{qnf}{p_{1}(mp_{1}+nq^{\prime})})^{\frac{1}{\alpha}}(k-\frac{qnf}{p_{1} (mp_{1}+nq^{\prime})})^{-\gamma}, \tag{22}\]  
Among them, \(\gamma=3-\frac{n(q+q^{\prime})}{mp_{1}+nq^{\prime}}\). It can be obtained from the above formula analysis that this is a single power law distribution form. This result also shows that the process of adding edges in the network is a necessary condition for the deviation of the power law distribution. In addition, through Formulas (14), (16) and (17), the expression of the curve turning point of the two-segment power law distribution is as follows:  
\[k_{c}=(cp_{2}t)^{\frac{mp_{1}+nq^{\prime}}{f}}(f+cp_{2}t)^{\frac{2f-mp_{1}-nq ^{\prime}}{f}}, \tag{23}\]  
As seen from the above formula, the value of the turning point is directly proportional to time. In areas where the degree value is smaller than this turning point, the network degree distribution is a stable distribution with a power exponent of \(\gamma_{1}=1.5\); and in areas greater than this turning point, the degree distribution follows the unsteady distribution characteristic of \(\gamma_{2}=3-\frac{n(q+q^{\prime})}{mp_{1}+nq^{\prime}}\), which also illustrates the long tail of the SCN. Some evolutionary situations have dynamic characteristics. * Probability of joining new edges \(p_{1}\). However, when there are no new companies to join the SCN, i.e., when \(p_{1}=0\), only the random growth and selective growth of the continuous side and the continuous reconnection process will be considered to affect the evolution of the SCN. Among them, \(p_{1}t\) in Formula (13) represents the scale of the SCN after time t evolves. Because \(p_{1}=0\), the network scale is the initial SCN scale m0. Based on continuity theory, the formula for the rate of change in the available network degree value with time can be expressed as:  
\[\frac{\partial k_{i}}{\partial t}=\frac{2 cp_{2}tk_{i}}{\sum\limits_{j}k_{j}}+ \frac{nqk_{i}}{\sum\limits_{j}k_{j}}-\frac{nq}{m_{0}}, \tag{24}\]The first item on the right side of the medium number in the above formula represents the process of joining the edges in the evolution of the network on the time scale, and the remaining items are the process of reconnecting the edges in the network.