# Analysis of Cascading Failures and Recovery in Freeway Network Under the Impact of Incidents
## 3 Methodology
### Recovery Strategies  
Cascading failure and recovery both fall within the research scope of complex network dynamics. Since cascading failures are unavoidable, studying response strategies after failures occur has practical significance. To better illustrate the research scope of this section, reference [14] presents the network failure and recovery evolution curves shown in Figure 3.  
When selecting failed components for recovery, limited resources prevent the restoration of all components at once, so only a portion of system can be chosen for recovery [44]. However, restoring some isolated nodes and edges does not effectively improve network performance, leading to a waste of recovery resources. Therefore, it is necessary to first define and identify the critical edges in the network, and then formulate a reasonable strategy for their recovery. \begin{table}
\begin{tabular}{l l} \hline \hline \multirow{3}{*}{Step} & Initialization of network parameters \\  & 0.1 Initialize the cascade process by setting \(t=0\). \\  & 0.2 Generate the structural representation of the freeway network. \\  & 0.3 Import the origin–destination (OD) traffic demand and implement the Method of Successive Averages (MSA) to estimate baseline link loads. \\ \hline \multirow{3}{*}{1} & Set the initial attack \\  & Remove the targeted links from the freeway network structure. \\  & Move cascade \(step=t+1\). \\  & Load the OD demand \(q(t)\) at cascade \(step\)\(t\). \\  & Update the link travel time function \(t_{a}\), if certain roads experience overload failures. \\  & Solve for \(D_{a}(t)\) using an SUE model at each cascade step \(t\), applying the MSA algorithm for stochastic loading. \\  & 5.0 Initialize the traffic assignment variable \(D_{a}^{(1)}(t)=0\), and set the iteration counter \(n=1\). \\  & 5.1 Update the travel time function for each network link. \\  & 5.2 Compute the auxiliary variable by solving an optimization problem. The result, denoted as \(y\), represents a feasible direction for descent in the solution space. \\  & 5.3 Update. Update the decision variable \(x\) using the following relationship: \(x_{i}^{k+1}=x_{i}^{k}+\frac{1}{k}\left(y_{i}^{k}-x_{i}^{k}\right)\). The coefficient \(\frac{1}{k}\) decreases with each iteration, reducing the magnitude of adjustments and stabilizing the flow over time. \\  & 5.4 Convergence criterion. The iteration continues until the difference between successive values of \(x\) falls \\  & below a defined gap threshold, ensuring that the flow stabilizes and fluctuations are minimized; otherwise, \\  & \(n=n+1\) and go to step 5.1. \\  & Overload conditions cause subsequent failures in previously unaffected links. \\  & Quantify the additional failure occurrences and analyze the associated performance measures of the network. \\  & Terminate the cascading failure process if no new failure links are generated; otherwise, return to step 2. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cascading failure procedure. 3.3.1. Identifying and Ranking Critical Road Segments  
Identifying critical links in traffic networks is a classical problem. Current research focuses on identifying critical road segments by analyzing and comparing the impact on network performance before and after segment failures, where the degree of impact equates to criticality. However, existing studies often do not effectively explore cascading failure recovery scenarios. This paper combines structural and functional indicators of road segments, determines the weight of each evaluation criterion using entropy values, and then ranks the road segments using the technique for order preference by similarity to the ideal solution (TOPSIS). Specifically, this paper considers the traffic demand and topology of road segments to construct a decision matrix \(X\) for evaluating the importance of road segments based on their volume, edge betweenness, and volume-to-capacity ratio (VCR).