# 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
Similarly, when the time step is ti, the expression that can be solved for the above formula is:  
\[k_{i}(t)=t^{\kappa}(m_{0}-\frac{qn^{2}(qt-q)}{m_{0}t_{i}^{\kappa-2}(m_{0}{}^{ \kappa}+nqt_{i})})+\frac{qn^{2}(qt-q)}{m_{0}t_{i}^{\kappa-2}(m_{0}{}^{\kappa}+ nqt_{i})}, \tag{25}\]  
When a new node is added at each time step \(t\), a node is randomly selected as an end node of the newly connected edge, and the other end node is selected according to Formula (2). During the evolution of the network, the total number of nodes remains unchanged, and the nodes with larger degrees in the network gradually increase, while the nodes with smaller degrees gradually decrease. After \(t\approx\) m02/4 time steps, the entire network will generate a fully connected graph, resulting in the final degree distribution of the SCN tending to \(\delta\). At this time, due to \(p_{1}=0\), the formula \(\gamma_{2}=3-\frac{n(q+q\prime)}{mp_{1}+nq\prime}\) can be expressed as \(\gamma_{2}=2-\frac{q}{q\prime}\), the turning point of the power law distribution \(k_{c}\
eq 0\), so the SCN is still a two-stage power law distribution, and the network nodes in the region with a larger degree still have a dynamic growth power index. At the same time, it also reflects that the joining process of new nodes will not change the distribution form of the node value of the entire SCN, but it has a certain effect on the deviation of the power law distribution of the network. * Link reconnection parameter \(n\)  
Similarly, analyze the evolution of the entire complex SCN when the process of reconnecting the edges of the SCN does not occur, i.e., \(n=0\). According to continuity theory:  
\[\frac{\partial k_{i}}{\partial t}=\frac{(mp_{1}+2 cp_{2}t)k_{i}}{\sum\limits_{ j}k_{j}}, \tag{26}\]  
The above formula shows the effect of the joining mechanism of nodes and edges on the rate of change of the node degree during the network evolution. Similarly, when the time step is \(t_{i}\), \(k_{i}(t)=m(\frac{t}{t_{i}})^{2}\) can be obtained. The probability density of the SCN is the same as that in Formula (19) and the power index \(\gamma_{1}=1.5\). It can be seen that the process of continuous reconnection will not affect the distribution of the nodes with a smaller value in the network, which is also in line with the characteristics of the optimal connection of the SCN, and at the same time verifies the effectiveness of the model in this paper. In addition, according to the formula \(\gamma_{2}\), when \(n=0\), \(\gamma_{2}=3\) is obtained. Therefore, when the reconnection process does not occur, the entire SCN follows a two-stage power law distribution and has a relatively stable evolution process. At this time, the two fixed power indexes of the node degree distribution are 1.5 and 3 [11] also pointed out that the two-segment power exponents of the node degree distribution of the agricultural product SCN are 2.78 and 1.78, respectively. Similar results can also be found in the human language network [18], whose two fixed power indexes are \(\gamma_{1}\)= 1.5 and \(\gamma_{2}\)= 2.7.