# Vulnerability Analysis of a Multilayer Logistics Network against Cascading Failure
## 4 Vulnerability Analysis  
### Load Vulnerability  
Based on the weighted centrality measurements, the top ten cities in different network layers with weighted centrality are shown in Table 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Metrics** & \(G_{a}\) & \(G_{b}\) & \(G_{c}\) & \(G_{d}\) & \(G\) \\ \hline Node & 274 & 496 & 528 & 138 & 918 \\ Average degree & 106.17 & 16.69 & 4.19 & 126.53 & 35.19 \\ Network diameter & 4 & 6 & 10 & 2 & 6 \\ Average clustering coefficient & 0.714 (0.378) & 0.687 (0.034) & 0.259 (0.007) & 0.927 (0.912) & 0.546 (0.038) \\ Average shortest path length & 1.875 (1.622) & 2.710 (2.517) & 3.698 (3.630) & 1.076 (1.087) & 2.550 (2.172) \\ Small world index & 0.381 & 0.254 & 0.070 & 0.862 & 0.214 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistical characteristics of networks. Figure 4: Fitting curves for the degree cumulation distribution of the multilayer network. The \(X\)-axis represents the capacity parameter, and the \(Y\)-axis represents the network efficiency. The capacity parameter increases from 0 to 1, with a step size of 0.1. The change in network efficiency with capacity parameters after cascade failure is analyzed using two node attack strategies: maximum load attack and minimum load attack. The simulation results of four single-network layers and comprehensive multilayer networks, namely, the maritime transport network, air transport network, logistics enterprise organization network, and international trade network, are given. As shown in Figure 5, (1) when the capacity parameter of the node is small, the network efficiency remains at a low state. With the increase in the capacity parameter of the node, the network efficiency is significantly improved. When the capacity parameters vary from 0 to 1, the network efficiency curve constantly decreases, and the reason for this is that with the increase in capacity parameters, the city's ability to bear the load will increase, the dynamic load distribution makes the load distribution uniform, cities will not be better able to bear the node load, and sensitivity to external disturbances will be reduced. When the capacity parameter increases to 0.5, the increase in capacity parameter enables the network efficiency to maintain a high state, leading to a reduction in network vulnerability and better connectivity performance of the urban network of logistics hubs. (2) Under different node attack strategies, the network efficiency is greatly different. In the case of a minimum load attack, due to the small load of failure node transfer, it is unable to cause a large-scale impact on other cities; that is, the network has strong resistance, and it is not easy to cause a cascading failure. For the maximum load attack, after the node with a larger load is removed, the original load will be distributed to other nodes in the network, and the intact node may be removed after receiving the load of the failed node, which may exceed its node capacity, thus causing the maximum loss of network efficiency. Even when the capacity parameter reaches the maximum, due to the increase in failure nodes, the connection of nodes in the network will decrease accordingly, and the realization of city interconnection requires longer links, which affects the network efficiency to some extent. Therefore, the nodes with high loads in the urban network of the logistics hub can maintain the efficient operation of the network, which should be protected to avoid serious damage to the network due to cascade failure.