# Vulnerability Analysis of a Multilayer Logistics Network against Cascading Failure
## 4 Vulnerability Analysis
### Load Vulnerability
\begin{table}
\begin{tabular}{p{142.3 pt} p{142.3 pt}} \hline
**Network Layer** & **Top 10 Cities in Terms of Weighted Centrality Values** \\ \hline Maritime Shipping Network & Shanghai, Ningbo, Singapore, Shenzhen, Busan, Hong Kong, Qingdao, Klang, Rotterdam, Kaohsiung \\ Worldwide Air-transportation Network & Frankfurt, London, Hong Kong, Munich, Vienna, Dubai, Memphis, Miami, Shanghai, Cincinnati \\ Logistics Enterprise Organization Network & Shanghai, Singapore, Hong Kong, Dubai, Chicago, Dallas, Shenzhen, Miami, Basel, Cincinnati \\ International Trade Network & Shanghai, Hong Kong, Busan, Rotterdam, Singapore, Frankfurt, Paris, Vancouver, Hamburg, Le Havre \\ Multilayer Logistics Network & Singapore, Shanghai, Hong Kong, Klang, Frankfurt, Shenzhen, Busan, Ningbo, Rotterdam, Miami \\ \hline \end{tabular}
\end{table} 
Table 3: Top 10 cities in terms of weighted centrality values in different network layers. The \(X\)-axis represents the capacity parameter, and the \(Y\)-axis represents the avalanche scale. The capacity parameter increases from 0 to 1, and the step size of proportional growth is 0.1. The avalanche scale actually reflects the degree of damage to nodes. Under different capacity parameters, the maximum load and the minimum load attack strategies are adopted to calculate the avalanche scale of the network when the cascade failure reaches a stable state. 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. It can be seen in Figure 6 that (1) when the capacity parameter of the node is small, the attack on the node with the maximum load will cause a large avalanche scale, which will cause part of the network to not run normally. With the increase in the capacity parameter, the avalanche scale will obviously decrease, and only a few node cities may encounter failures. For example, when the capacity parameter \(\alpha=0.1\), the avalanche scale generated by the attack on the highest load node is more than 50%. The flow of each node in the network is close to the city's maximum capacity limit, which leads to the city's high sensitivity. Figure 5: The network efficiency of different attack strategies varies with capacity parameters under cascade failure. The disturbance will cause more nodes to be destroyed, leading to a large-scale cascade effect. When the capacity parameter \(\alpha>0.5\), the failure scale of the overload cascade gradually decreases, and the avalanche scale converges to 0, indicating that the network cannot trigger cascade failure or that only a few nodes can fail, which conforms to the rule that cascade failure damage decreases with increasing node capacity. Therefore, larger capacity parameters can avoid or slow down the occurrence of cascading failures. (2) Under different node attack strategies, the avalanche scale varies greatly. The attack method of removing the highest load causes more cascading failure nodes in the network than the lowest load, and the size of failure nodes generated is 0-40%. The avalanche scale curve generated by the attack on the lowest load node presents a curve parallel to the \(X\)-axis and infinitely approaching 0, indicating that under the attack mode of the lowest load, all the nodes except the attacked node can maintain the existing logistics function. Figure 6: The avalanche size of different attack strategies varies with capacity parameters under cascade failure.