# Identification of Critical Road Links Based on Static and Dynamic Features [PERSON]
## 3 Methodology
### Traffic Link Representation Learning  
After constructing the directed dual traffic network \(G\), various indicators can be calculated to evaluate the importance of links. However, the extraction of importance indicators, such as adjacency matrix, degree, betweenness centrality, and network efficiency, from complex networks is a time-consuming process. Additionally, as the size of the network increases, the feature matrix obtained from these methods exhibits high dimensionality and sparsity, rendering clustering-based evaluations of link importance barely satisfactory. Graph representation learning (GRL) can address these limitations well, which is a technique that enables the extraction of effective features from graph information. One model that is widely used for feature extraction in GRL is Node2 vec. Based on the Deepwalk model [21], Node2 vec [22] integrates walk bias, which incorporates depth-first search (DFS) and breadth-first search (BFS) balance coefficients to consider both local and global features of nodes. Node2 vec, introduced in 2016, takes the network structure as input and outputs the embedding vector of each node. Its objective is to map the node information in the network to a low-dimensional, continuous, and dense feature space. The embedding vectors both preserve the neighborhood features of the original node and facilitate subsequent data mining. In this paper, we propose a novel representation learning model named TraLink2 vec, which extends the Node2 vec method by integrating both static and dynamic features in the traffic network. TraLink2 vec inherits the walk bias of Node2 vec for the static road network structure from the perspectives of BFS and DFS, while also incorporating the dynamic travel features of the traffic system to construct dynamic travel bias from the perspective of traffic demand. To sample the directed dual traffic network \(G\), TrLink2 vec simulates a random walk process of fixed length \(l\) for \(r\) times. Given a current node \(c_{i-1}\), the next node \(c_{i}\) is generated based on the following distribution:  
\[P(c_{i}=x\mid c_{i-1}=v)=\left\{ \begin{array}{ll}\frac{\pi_{vx}}{2}&\text{if }(v,x)\in E\\ 0&\text{otherwise}\end{array} \right. \tag{2}\]  
where \(\pi_{vx}\) is the unnormalized transition probability between nodes \(v\) and \(x\), and \(Z\) is the normalization constant. In order to balance the dynamic and static features in the traffic network, we use a second-order random walk procedure to guide the walk. Suppose the current random walk passes through the edge \((t,v)\) to reach \(v\). We set the unnormalized transition probability \(\pi_{vx}=\pi(x\mid t,v)\) as follows:  
\[\pi(x\mid t,v)=\left\{ \begin{array}{ll}\alpha\times\frac{1}{p}\times w_{vx} +\beta\times\frac{\varepsilon_{flow,x}}{\varepsilon_{spot,x}}&\text{if }d_{tx}=0\\ \alpha\times w_{vx}+\beta\times\frac{\varepsilon_{flow,x}}{\varepsilon_{spot,x}}&\text{if }d_{tx}=1\\ \alpha\times\frac{1}{\alpha}\times w_{vx}+\beta\times\frac{\varepsilon_{flow, x}}{\varepsilon_{spot,x}}&\text{if }d_{tx}=2\end{array} \right. \tag{3}\]  
Figure 1: Example of directed dual traffic network. \[\varepsilon_{flow\_x}=\frac{flow_{x}}{flow_{v}} \tag{4}\]  
\[\varepsilon_{speed\_x}=\frac{speed_{design}}{speed_{x}} \tag{5}\]  
where \(\pi(x\mid t,v)\) is the unnormalized transition probability of the next link node \(x\) through the edge \((t,v)\). \(d_{tx}\) is the shortest path distance between link node \(t\) and link node \(x\). \(w_{vx}\) is the weight between link node \(v\) and link node \(x\), which means the length between the midpoints of the two link nodes. Parameter \(p\) is the return parameter, which controls the probability of returning to the previous link node. Parameter \(q\) is an in-out parameter, which controls the probability of returning to the previous link node or exploring far-away link nodes. Parameter \(\alpha\) is the control weight of static features and parameter \(\beta\) is the control weight of dynamic features.