# Analysis of Cascading Failures and Recovery in Freeway Network Under the Impact of Incidents
## 3 Methodology
### Modeling the Cascading Failures of the Freeway Network
#### 3.2.2 Load Redistribution  
To better integrate complex network theory with transportation systems, this section introduces the traffic flow assignment method under cascading failures. The route choice behavior of travelers can be described using a probabilistic model. The Logit stochastic path choice model is a widely adopted assignment approach that reflects drivers' preferences for routes they subjectively perceive to have the lowest impedance, denoted as \(C_{k}^{ij}\). Assuming that the minimum impedance of the path between the origin and destination is represented by \(k\), the value of \(C_{k}^{ij}\) can be calculated using Equation (4), and \(C_{k}^{ij}\) represents the actual impedance of the path, while \(\lambda_{k}^{rs}\) denotes the difference between subjective perception and actual path impedance (\(\lambda_{k}^{ij}\) represents the random error component, \(E\Big{(}\lambda_{k}^{ij}\Big{)}=0\)). \[C_{k}^{ij}=c_{k}^{ij}+\lambda_{k}^{ij}\,\forall k,i,j \tag{4}\]  
The traditional Logit model calculates path choice probabilities under the assumption that each path is independent, primarily based on the total cost of each path. This assumption ignores the overlapping portions of the paths, which may lead to excessive allocation of traffic to these overlapping paths. This means that although certain paths share many of the same segments, the model does not adjust the choice probabilities of these paths to reflect their actual similarity, which does not align with real-world traveler behavior. Consequently, this allocation algorithm may excessively assign traffic flow to overlapping paths, indicating that it is insensitive to the network structure and potentially resulting in more severe consequences. Therefore, during the cascading failure process, we use the C-Logit model: by introducing a path overlap correction factor, we adjust the path choice probabilities to reflect the actual similarity between paths. The C-Logit model introduces a path overlap correction factor (C-factor) to reduce the choice probabilities of overlapping paths. This correction factor reflects the degree of overlap between paths, making the choice probabilities of paths with more overlapping segments relatively lower. The path choice probability in the model is as follows:  
\[p_{ij}^{k}=\frac{\exp\left[-\theta\left(c_{ij}^{k}+\lambda lnS_{k}\right)\right. \right.}{\sum_{l\in P_{ij}}\exp\left[-\theta\left(c_{ij}^{l}+\lambda lnS_{l} \right)\right.} \tag{5}\]  
where \(S_{k}\) is the path size factor for path \(k\), reflecting the relative length and overlap of the path, and \(\lambda\) is an adjustment parameter used to control the influence of the correction factor. \[S_{k}=\sum_{a\in k}\frac{l_{a}}{L_{k}}\left(\frac{1}{\sum_{j\in C}\delta_{aj}}\right) \tag{6}\]  
where \(l_{a}\) denotes the length of edge a, and \(L_{k}\) denotes the length of path \(k\). The term \(\sum_{j\in C}\delta_{aj}\) represents the number of paths containing the edge \(a\). During cascading failures, the travel time on unaffected road segments is influenced by both their capacity \(C_{a}\) and the corresponding flow demand \(D_{a}(t)\). The Bureau of Public Roads (BPR) function is commonly applied to represent this dependency, where the travel time on a given link \(i\) increases as the demand-to-capacity ratio rises. \[t_{a}=t_{0}[1+\alpha\cdot(\frac{D_{a}}{C_{a}})^{\beta}] \tag{7}\]  
Here, \(t_{a}\) denotes the average travel time per unit distance on link \(a\) under normal traffic conditions, while \(t_{0}\) represents the corresponding free-flow travel time. \(D_{a}\) and \(C_{a}\) refer to the traffic demand and the capacity of the link, respectively. The parameter \(\alpha\) controls the proportion between free-flow travel time and travel time at full capacity, whereas \(\beta\) governs the sensitivity of travel time to increases in demand. Empirical studies commonly adopt \(\alpha=0.15\) and \(\beta=4\), which are the standard values recommended in official guidelines.