# 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 SCN Based on Small-World Network Evolution  
In the early days, a type of complex SCN that was studied more internationally was a random network. In an ideal state, when the scale of a random SCN gradually tends to infinity, the degree distribution of the network will obey the Poisson distribution [11], as follows:  
\[P(k)=e^{-pN}\frac{\left(pN\right)^{k}}{k!}=e^{-\left\langle k\right\rangle} \frac{\left\langle k\right\rangle k}{k!}, \tag{1}\]  
where \(\left\langle k\right\rangle\) is the average value of the node degree of the SCN. Usually, the number of nodes in a random SCN will peak at a node degree value of \(\left\langle k\right\rangle\), and then the number of nodes will decay exponentially.  
The E-R model in the random SCN is a network distribution model in which the degrees of vertices obey the binomial distribution, namely \(P(k)=C_{n-1}^{k}p^{k}(1-p)^{n-1-k}\) and the basic idea is to connect each pair of nodes in the \(N\) nodes with probability \(p\). However, the two problems in the E-R model are \(p\) is a constant in the model, but in reality, \(p\) and \(n\) are inversely proportional; the mean value of the network nodes is linearly related to \(n\), but the actual mean growth rate will be slower. A schematic diagram of the E-R stochastic SCN is shown in Figure 1a. Its network node \(N=20\) and node connection probability \(p=0.1\).  
However, with the deepening of scientific research, many of the complex SCNs observed in reality do not simply meet the new edge to join with the same probability; the network's agglomeration coefficient and degree value have significant heterogeneity, rather than Obey the uniformity and homogeneity network characteristics with an agglomeration coefficient of \(C_{ER}\approx\frac{\left\langle k\right\rangle}{N}\). To further study the network topology and evolution mechanism between a completely regular network and a random network, [PERSON] and [PERSON] of Cornell University proposed the concept and model of a WS small-world network in 1998.  
In reality, many complex SCNs exhibit the \"small-world\" characteristics of clusters; that is, any SCN node can connect with nodes on any other network in a few simple steps, which is unique to complex SCNs. Figure 1b shows a schematic diagram of a WS small-world SCN with 20 nodes, each node having four neighbors, and a random edge reconnection probability of 0.3. The main idea of its construction is based on a node with \(N\) nodes, and each node has a closed regular network of k-nearest neighbors that interrupts and reconnects the connected edges in the network with a certain probability. The specific construction process of the WS small-world SCN is shown in Figure 2.  
The SCN conforming to the \"small-world\" characteristics has a clustering coefficient of \(C(0)\approx(\text{k}-1)/\text{k}\) similar to the regular network and a smaller average path length \(dist_{c}\propto\log(N)\) similar to the random SCN. In the evolution of complex SCNs over time, for  
Figure 1: (**a**) E-R stochastic supply chain network; (**b**) WS small-world network.  
Figure 2: WS small-world network construction flow chart.  
a small reconnection probability p, the network aggregation coefficient can be large, but the change is small, and the average network path length can be small, but the attenuation is large [12].