# Evolution of the Complex Supply Chain Network Based on Deviation from the Power-Law Distribution
## 2 Evolutionary Characteristics and Model Analysis of Complex SCNs
### SCN Analysis Based on Scale-Free Network Evolution  
Based on research on the evolution of small-world SCNs, many experts have found that the topological structure characteristics of most complex SCNs are different from those of stochastic SCNs and WS small-world supply chains when they conduct further research on real networks in different fields. For the network, the most significant difference is the degree distribution of the network. It can be seen from the previous section that the more average the degree value of the random SCN is, the more concentrated the degree distribution interval; there are also fewer nodes with larger deviations from the mean value of the degree, and it decays exponentially for the larger degree. The exponential form rises to the apex and then decays to the exponential form, approximately obeying the Poisson distribution, the degree values of most nodes are gathered around a certain value, and the number of other degree values is relatively small. However, in most realistic networks, such as complex SCNs and complex aviation networks, etc., the terminal on the right side of the degree distribution curve has the characteristic of a \"fat tail\", indicating that such SCNs have certain nodes with large degree values; that is, the network existing nodes have a large number of connected edges and are highly nonuniform, unlike random SCNs and small-world SCNs. At present, the observed degree distributions of most realistic and complex networks have power-law characteristics. Because power laws have the same properties on all scales, such networks are called scale-free networks. Dynamic growth and preferential connection are two evolutionary mechanisms for forming scale-free networks. To further analyze the topology structure and network degree distribution characteristics of the scale-free SCN, Python software was used to generate a BA scale-free network with node \(n=500\) and the initial number of neighbors \(m=1\) for each node. As shown in Figure 3, it can be seen that (a) that the network structure is unevenly distributed, with a large number of nodes with small degree values, but there are also a few core nodes with large degree values, and the network node degree value is [1; 37]. The distribution of the network degree of the logarithmic transformation of the ordinate is shown in (b), where the discrete points of the degree value are fitted. The coefficient R2 of the linear fitting function is 0.91, which has a good fit, so it shows that the scale-free SCN follows the power-law distribution feature. Figure 3: BA scale-free network simulation diagram: (**a**) Network topology; (**b**) Network degree distribution fitting. For a complex SCN under economic globalization and an open market environment, each node company's position and structural characteristics in the network are different, and this will be due to the participation and cooperation between nodes or supply chains. Eliminate network nodes, and node enterprises develop the continuous evolution of the SCN with a nonuniform growth model under the motivation of survival and maximum profit. Through research and analysis, the complex SCN has the following three characteristics:  
* Growth: Assuming that the initial SCN G0 (n0, l0) has n0 nodes and l0 connected edges, from time t0, a new node with \(m\) edges is added in an exponential form \(N(t)\propto e^{c_{n}t}\), and the new node will have a certain probability of connecting existing nodes in the network. * Optimal connectivity: When new nodes join the SCN, they will generally prefer to select larger-scale, higher-degree node companies to connect; that is, the probability of selecting a connected node \(\prod\limits_{j\to i}\) is proportional to the node degree value \(k_{i}\).