# Vulnerability Analysis of a Multilayer Logistics Network against Cascading Failure
## 2 Methodology and Data
### The Model
The International Trade Network is characterized by its complex structure, reflecting the scale of trade and the diversity of economic activities. The existence of these four layers in a city network acknowledges the complexity of modern urban systems and the need for an integrated approach to understanding urban dynamics. Each layer contributes to the overall connectivity and functionality of the city network, enabling it to respond effectively to the demands of globalization. Understanding the interactions between these layers is essential for effective urban planning, logistics development, and the enhancement of city competitiveness. An integrated logistics hub city includes at least two attributes of a single-factor network, such as a combination of port-based and air-based logistics hub cities. Since the role of a logistics hub city network diffuses through city connections, this paper emphasizes not only the hub facilities radiating from routes in the transportation network, such as airports and seaports but also the placement of logistics firms in the city and the city's trade-generating capacity. The trade generating capacity reflects the intercity trade relations, the size and function of the logistics hub city itself, and its location, while the location of hub facilities and logistics firms reflects the city selection preferences of carriers in the logistics transportation system to emphasize the additional level of activity that geographic location conveys to the logistics hub city. Therefore, the networks characterized by maritime transport, air transport, logistics company organization, and international trade flows are constructed from three domains: a physical network, organizational network, and trade network, respectively, to serve trade and mobility through different functional roles. The integrated network is considered a logistics system consisting of interdependent and complementary subnetwork layers serving international trade and goods flows, and each subnetwork layer is considered an independent system or network. The analysis of the global logistics network is considered a four-dimensional configuration. \(G_{\Pi}(\Pi=a,b,c,d)\) is defined as four different kinds of single layers, which are closed contact through co-owned cities. The principle of construction is as follows: four single layers \(G_{a}\), \(G_{b}\), \(G_{c}\) and \(G_{d}\) of a multilayer network \(M(G=\{G_{\Pi},\Pi\in\mathbb{N}\},C)\), the transformation of the form \(g_{x}:G_{i}\to G\), where \(x\) is regarded as a given topological metric, such as degree, \(G\) is the family of the layers, and \(C\) is the connections between different layers. In this context, the set \(T_{G}=\{g_{X}:X\}\)\(X\) is defined as the attribute of the network, which represents the effects of network topology caused by the node converging. The structure of single layers is modeled as a graph of \(N_{\Pi}\) nodes. The single layer \(G_{\Pi}=(V_{\Pi},E_{\Pi},W_{\Pi})\) is composed of a set of city nodes \(V_{\Pi}\) and a set of edges \(E_{\Pi}\) between them. The weighted matrix \(W_{\Pi}\) is defined as follows. \[W= \begin{bmatrix}0&W_{12}&\cdots&W_{1(n-1)}&W_{1n}\\ W_{21}&0&\cdots&W_{2(n-1)}&W_{2n}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ W_{(n-1)1}&W_{(n-1)2}&\cdots&0&W_{(n-1)n}\\ W_{n1}&W_{n2}&\cdots&W_{n(n-1)}&0\end{bmatrix} \]  
The four single layers are characterized by a maritime network, air transportation network, logistics enterprise organization, and international trade network, as shown in Table 1. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Mark** & **Name** & **Notes** \\ \hline \(G_{a}\) & Maritime Shipping Network, MSN & Container liner route to the port city \\ \(G_{b}\) & Worldwide Air-transportation Network, WAN & Air cargo transportation directly to the airport city \\ \(G_{c}\) & Logistics Enterprise Organization Network, LEN & Logistics company headquarters and branch city \\ \(G_{d}\) & International Trade Network, ITN & International trade import and export city \\ \hline \hline \end{tabular}
\end{table}