# Analysis of Cascading Failures and Recovery in Freeway Network Under the Impact of Incidents
## 3 Methodology
### Evaluation Metrics  
In the evolution of network cascading failures and recovery, this paper introduces four metrics to capture the changes in network performance following external attacks and recovery, illustrating the extent of damage caused by cascading failures and the subsequent changes post-recovery. These metrics are employed to numerically simulate the cascading failure process in the freeway network and quantify the resulting impact.  
Network Connectivity: The largest connected subgraph can be used to describe the connectivity of freeway networks. The larger the largest connected subgraph, the more stable the freeway network is, and the stronger its ability to resist attacks or interference; conversely, the weaker the freeway network is. When the nodes or edges of the network are disrupted, the network splits into several sub-networks, and the sub-network with the most nodes is the largest connected subgraph G, which can be expressed as  
\[G=\frac{N^{\prime}}{N} \tag{13}\]  
where \(N^{\prime}\) represents the total number of nodes in the largest connected subgraph.  
Network Robustness: The proportion of failed edges quantifies the ratio of failed edges to the total number of edges during cascading failures in the freeway network. The calculation formula is as follows:  
\[P_{f}=\frac{E_{f}}{E_{t}} \tag{14}\]  
where \(P_{f}\) denotes the proportion of failed edges, \(E_{f}\) represents the number of failed edges, and \(E_{t}\) is the total number of edges in the network.  
This formula calculates the proportion of edges that have failed at a given moment relative to the total number of edges in the entire network. It is commonly used to analyze the extent of damage to the network during cascading failures and to evaluate the network's robustness.  
Network Efficiency: Network efficiency \(E\) is an indicator used to measure the overall efficiency of a network, particularly in the context of potential disruptions to nodes or edges. This metric reflects the ease of volume transmission between any two nodes within the network. The formula for calculating network efficiency is as follows:  
\[E=\frac{1}{N(N-1)}{\sum_{i\
eq j}\frac{1}{d_{ij}}} \tag{15}\]  
where \(N\) represents the total number of nodes in the network, and \(d_{ij}\) represents the shortest path between nodes. \(i\
eq j\) indicates that each pair of nodes (i.e., two different nodes) is considered once in the summation.  
In a freeway network, the essence of this formula is the averaging of the reciprocals of the shortest paths between all pairs of nodes. This implies that when the shortest path between two nodes is shorter, the efficiency between them is higher, which, in turn, increases the overall efficiency of the network. Conversely, if the shortest paths between most node pairs are longer, the network's efficiency will decrease.  
Network Connective Reliability: Connectivity reliability in a freeway network is measured as the total probability that at least one valid route exists between each origin-destination pair [45]. This indicator is influenced by factors such as the shortest path distances, the number of available paths, and the overall scale of the network. It can be mathematically represented as  
\[CR=\frac{2}{(N-1)}\sum_{i,j\in N}^{i\
eq j}\sigma_{ij} \tag{16}\]  
in which  
\[cr_{ij}=\frac{N_{ij}^{\prime}d_{ij}}{N_{ij}d_{ij}^{\prime}} \tag{17}\]  
Here, \(cr_{ij}\) denotes the connectivity reliability between nodes \(i\) and \(j\). \(N\) refers to the total number of nodes in the network. \(N_{ij}^{\prime}\) indicates the number of viable paths between \(i\) and \(j\) after a set of deletion-based attacks, while \(N_{ij}\) corresponds to the number of such paths under normal, unperturbed conditions. The summation is performed over all unique node pairs within the network, totaling \(N(N-1)\) combinations.