## 4 Case Analysis
### Simulation Attack and Performance Analysis  
#### 4.4.1 Network Performance Indicators  
Network performance indicators reflect the network's operational capabilities. These indicators tend to decline as nodes or edges fail, as reflected in the network's robustness, reliability, resistance to destruction, and vulnerability. While these indicators are independent, they complement each other and are somewhat interconnected. Therefore, studying the changes in these key indicators is crucial for measuring the performance of complex networks. Robustness, vulnerability, and resistance to destruction: robustness indicates the fault tolerance of the metro network when facing random attacks; vulnerability reflects the survivability of the transport network when targeted, showing the fault tolerance and survival of the stations and the overall network; destruction resistance is the ability of a network to maintain or restore its performance to an acceptable level when deterministic or random failures occur. Network attack: This often involves simulating random or deliberate attacks to observe the overall network performance's decline after an attack. Random attacks refer to unforeseen major incidents in the metro network, such as equipment failures, force majeure, severe weather conditions, etc. ; deliberate attacks are the purposeful human destruction of key nodes. After a node is damaged, adjacent relationships are severed, resulting in isolated points. The metrics commonly used to describe network performance after node damage include network efficiency, network connectivity, maximum connectivity subgraph, and natural connectivity. Among them, network efficiency and connectivity reflect the utility and connectivity of the network, while maximum connectivity subgraph and natural connectivity reflect the damage resistance and overall operational performance of the network [34]. Network connectivity is the ratio of the number of connected edges \(l_{G}\) formed by the remaining nodes when some nodes in the network fail to the number of connected edges  
Figure 5: Node importance topology diagram (Using Gephi v0.9.2 with the latitude and longitude from Google Maps). \(l_{G}\) when the network is intact. The larger the value, the more complete the mesh formed by the remaining nodes and the smaller the network performance loss. The formula can be written as follows:  
\[\varphi=\frac{l_{G}}{l_{G\text{max}}}. \tag{20}\]  
Network efficiency reflects the transmission efficiency of the entire network. The higher the value of network efficiency, the lower the transmission difficulty and the higher the transmission efficiency, which can be used to measure network performance. The formula can be written as:  
\[E(G)=\frac{1}{N(N-1)}{\sum}_{i\
eq j\
eq G}\frac{1}{d_{ij}}. \tag{21}\]  
The size of the maximum connected subgraph is the number of nodes in the subgraph that has the highest number of nodes among the various connected subgraphs decomposed from the network. It primarily measures the network's resistance to damage. The larger the value, the more complete the network formed by the remaining nodes and the smaller the network performance loss. The formula can be written as:  
\[J=\frac{N^{\prime}}{N}, \tag{22}\]  
where \(N\) and \(N^{\prime}\) denote the size of the maximum connected subgraph before and after node damage, i.e., the number of nodes in the maximum connected subgraph formed by the network. The coal-rail-water intermodal network has realized a fully connected subgraph, so the initial \(N\) is 84. The relative maximum connectivity subgraph aims to explain network performance in terms of the network topology level. Since it's a ratio, it can directly characterize the change in network performance.