# 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
### Establishment of an Evolution Model of Power-Law Distribution Deviation in SCN  
Based on the analysis of the above model assumptions, each step of the evolution of the SCN can be expressed using a continuous theoretical model. The specific steps are as follows:  
* Add a new node to the network with probability \(p_{1}\); then, the degree change rate of the node is expressed as follows: \[\left(\frac{\partial k_{i}}{\partial t_{1}}\right)=\frac{\lambda p_{1}}{\sum \limits_{j}k_{j}},\] (10) We can know \(\lambda=m\). * Adding ct new connected edges in the network with probability \(p_{2}\), the rate of change in the degree of the node is: \[\left(\frac{\partial k_{i}}{\partial t_{2}}\right)=\frac{\mu p_{2}k_{i}}{\sum \limits_{j}k_{j}},\] (11) Among them, after each time step, the overall change in SCN connectivity is \(\Delta k=2 ct\), so \(\mu=2 ct\) is available. * When \(n\) consecutive edges in the network are reconnected, the degree change rate of the node can be expressed as: \[\left(\frac{\partial k_{i}}{\partial t_{3}}\right)=-\frac{\omega q}{p_{1}t_{3} }+\frac{\omega qtk_{i}}{\sum\limits_{j}k_{j}},\] (12)  
Among them, \(p_{1}t_{3}\) represents the scale of the SCN at time \(t_{3}\). In Formula (12), the first term on the right side of the equal sign is the reduction of the network node degree value caused by the connected edge with the probability q removed, and the second term expresses that the connected edges are reconnected with the optimal choice. The increase is in the network degree value. However, the total connectivity of the SCN did not change during the process of continuous reorganization \(\omega=n\). Since the above three changes will occur synchronously in the overall process of SCN evolution, the situation of the network node degree value change rate caused by the three evolution processes can be considered at the same time. The formula is as follows:  
\[\left(\frac{\partial k_{i}}{\partial t}\right)=\frac{(mp_{1}+2 cp_{2}t+nqt)k_{i }}{\sum\limits_{j}k_{j}}-\frac{nq}{p_{1}t}, \tag{13}\]  
Among them, the total value of the network is \(\sum\limits_{j}k_{j}=t(2 mp_{1}-nq+nqt)+cp_{2}t^{2}\). For the node that joined at time \(t_{i}\), \(k_{i}(t_{i})=m\). By iteratively solving Equation (13), the expression of the solution is as follows:  
\[ \begin{split}& k_{i}(t)=m(\frac{t}{t_{i}})^{\alpha}\left(\frac{f+cp_{2}t }{f+cp_{2}t_{i}}\right)^{\beta}-t^{\alpha}(f+cp_{2}t)\beta\\ &\times\frac{nqt^{-\alpha}(f+cp_{2}t)^{\alpha-1}(-f+mp_{1}-cp_{ 2}t+nq\prime)-nqt_{i}^{-\alpha}(f+cp_{2}t_{i})^{\alpha-1}(-f+mp_{1}-cp_{2}t_{i }+nq\prime)}{(f-mp_{1}-nq\prime)(nqt+mp_{1})}\end{split}  \tag{14}\]Among them  
\[\left\{ \begin{array}{l}\alpha=\dfrac{mp_{1}+nqt}{f}\\ \\ \beta=\dfrac{2f-(mp_{1}+nqt)}{f}\\ \\ f=2 mp_{1}-nq+nqt\end{array} \right. \tag{15}\]  
Equation (14) expresses the situation where the power-law distribution deviates in two different regions. At time \(t\), the degree \(k_{i}(t)\) of the node is less than the probability \(p(k_{i}(t)<k)=P(t_{i}>F(m,n,p_{1},q,qt,c,t)t)\) of \(k\). When \(t_{i}\ll t\), the expression of the degree \(k_{i}(t)\) of the network can be obtained according to continuity theory as follows:  
\[k_{i}(t)\approx m(\dfrac{t}{t_{i}})^{\alpha}\left(\dfrac{cp_{2}t}{f}\right)^{ \beta}+\dfrac{nq(cp_{2}t)^{2}}{(f-mp_{1}-nqt)(nq+mp_{1})} \tag{16}\]  
\[W(m,n,p_{1},q,qt,c,t)=m^{\frac{1}{\alpha}}(\dfrac{cp_{2}t}{f})^{\frac{2f-mp_{ 1}-nqt}{mp_{1}+nqt}}(k-\dfrac{nq(cp_{2}t)^{2}}{(f-mp_{1}-nqt)(mp_{1}+nqt)})^{ \frac{-1}{\alpha}} \tag{17}\]  
Assuming that the addition of nodes and edges and the reconnection of edges are completed in the same time period, the probability density of \(t_{i}\) is \(p_{i}(t_{i})=\frac{1}{t}\), and there is \(p(k_{i}<k)=1-W(m,n,p_{1},q,qt,c,t)\). The probability density \(p(k)\) at time \(t\) can be obtained by the following formula:  
\[