# Vulnerability Analysis of a Multilayer Logistics Network against Cascading Failure
## 2 Methodology and Data
### Topological Metrics  
Network efficiency is used to analyze the performance of the network, which could capture the difficulty of reaching the distant part of the network and the ability to share the flow and information. Improving network efficiency is a potential way to improve network reliability in reality. The vulnerability is associated with an attack and can be measured by a reduction in efficiency after that. Network efficiency is an important evaluation index of the network operation effect, reflecting the ease of materials from a specific city to reach the network farther away, used to measure the efficiency of the exchange of materials between cities. In practice, container liner companies or air cargo companies choose to open routes by not only considering the geographical distance between cities but also taking into account the demand for cargo transportation, the number of empty containers, transit costs, transportation conditions, and other factors. Therefore, instead of expressing the transportation distance between logistics hub cities in terms of traditional geographical distance, the minimum number of transit times connecting two logistics hub cities is used, i.e., the number of sides to be passed by the shortest path between the two cities is used to express the distance of city \(i\) and \(j\). Because the efficiency of transmission between the two logistics hub cities \(\epsilon_{ij}\) is  
Figure 2: Iterative steps for cascading failure. inversely proportional to the distance of the shortcut, \(\epsilon_{ij}=1/d_{ij}\) when \(d_{ij}=\infty\), \(\epsilon_{ij}=0\)[47]. The network efficiency is defined as:  
\[E=\frac{1}{n(n-1)}{\sum_{i\
eq j}}\epsilon_{ij}=\frac{1}{n(n-1)}{\sum_{i\
eq j }}\frac{1}{d_{ij}} \tag{7}\]  
where \(n\) is the total number of cities in the network and \(d_{ij}\) is a shortcut distance between cities. The inverse of the distance is set as the transmission efficiency between two cities, and the network efficiency is expressed as the average of the transmission efficiency between any two logistics hub cities in the network. If the shortcut distance is too large, the network efficiency is lower, which will affect the transmission of information and the flow of elements in the network, in which the transmission efficiency and diffusion of logistics hub cities are weaker, and the response speed to external disturbances lags and slows. In contrast, the smaller the shortcut distance is, the higher the network efficiency and the stronger the global connectivity and transmission performance of the network. The logistics hub city can realize the flow of goods at a relatively low cost and relatively fast, and it can enhance the resistance of the logistics hub city to external risks. Avalanche size is a measure of the degree of damage to network efficiency under the city node disruption scenario and is the proportion of the number of failures of a particular logistics hub city until the end of the cascade failure process will cause the failure of other nodes in the network to the number of cities in the initial state, defining the avalanche size as  
\[\delta=1-\frac{N^{*}}{N} \tag{8}\]  
where \(N\) and \(N^{*}\) are the number of cities in the network that can maintain normal operation before and after the cascade failure phenomenon. The larger the avalanche, the greater the damage to the overall network of logistics hub cities. Different topological metrics describe a specific structural characteristic of the global logistics network. In this work, we consider the following metrics. The degree \(k\) of a node is the number of edges linked with the others in a network [48]. \[k=\sum_{j=1,j\
eq i}^{n}X_{ij} \tag{9}\]  
where \(X_{ij}\) is the number of links between node \(i\) and node \(j\), and \(n\) is the total number of cities. The weighted degree \(k_{w}\) is an extension of the degree \(k\). \[k_{w}=\sum_{j=1,j\
eq i}^{n}w_{ij} \tag{10}\]  
where \(w_{ij}\) is the weight of the links of the node \(i\)[49].