# An Arch-Bridge Topology-Based Expressway Network Structure and Automatic Generation
## 3 Methodology
### Arch-Bridge Topology
_There are pairs of paths:_ \(g_{1}^{en},g_{1}^{ur},\ldots,g_{n-1}^{ur},g_{n}^{ex}\) _and_ \(g_{n}^{en},g_{n-1}^{dr},\ldots,g_{1}^{dr},g_{1}^{ex}\)_._  
_Here,_ \(g_{i}^{ur}\) _and_ \(g_{i}^{dr}\) _respectively represent the gantry nodes deployed on the upline and downline lanes of the road, also known as mutually exclusive gantry pairs, denoted as \(\left(g_{i}^{ur}\Leftrightarrow g_{i}^{dr}\right).\)_  
_As shown in Figure 2, this topology shape is similar to an arch-bridge; thus, it is named \"arch-bridge topology.\" The two adjacent toll stations and their intermediate ETC gantries on the upline and downline sections can form a minimal arch-bridge topology structure. Of course, the complex expressway network structure can be decomposed into simple arch-bridge topologies. Compared with traditional road network maps, the arch-bridge topology structure not only preserves the overall shape of the road network but also simplifies the complex road network structure, making the reconstruction of the entire expressway network traceable. To meet the needs of subsequent research, the following definition is given:_  
Figure 2: Arch-bridge topology. **Definition 2**. : _Node Degree: The number of gantry nodes directly adjacent to the gantry node \(g_{i}\) is called the degree of the gantry node \(g_{i}\), denoted as \(\pi(g_{i})\). Specifically, the number of gantry nodes directly connecting upline and downline to \(g_{i}\) on the road are referred to as the in-degree \(\pi_{in}(g_{i})\) and out-degree \(\pi_{out}(g_{i})\) of \(g\), respectively._  
\[\pi(g_{i})=\pi_{in}(g_{i})+\pi_{out}(g_{i}) \tag{1}\]  
Of course, for gantry station entrances (including provincial border entrance gantries), the gantry node \(g_{i}\) has \(\pi_{in}(g_{i})=0,\pi_{out}(g_{i})\geq 1\), \(g_{i}\in G_{en}\); for gantry station exits (including provincial border exit gantries), the gantry node \(g_{j}\) has \(\pi_{in}(g_{j})\geq 1\), \(\pi_{out}(g_{j})=0,g_{j}\in G_{ex}\); the middle gantry \(g_{k}\) has both in-degree and out-degree and \(\pi_{in}(g_{k})\geq 1\), \(\pi_{out}(g_{k})\geq 1\), \(g_{k}\in G_{r}\). According to the definition, given a mutually exclusive gantry pair \(\left(g_{i}\Leftrightarrow g_{j}\right)\), the in-degree and out-degree will have the following relationship:  
\[\pi_{in}(g_{i})=\pi_{out}\left(g_{j}\right) \tag{2}\]  
\[\pi_{out}(g_{i})=\pi_{in}\left(g_{j}\right) \tag{3}\]  
**Definition 3**. : _Directed edge \(e_{k}=\left\langle g_{i},g_{j}\right\rangle\in\mathbf{E}\), formed by two nodes \(\left(g_{i},g_{j}\right)\), has a distance denoted as the edge distance \(d(e_{k})\), which refers to the actual shortest path distance from node \(g_{i}\) to node \(g_{j}\)._  
Edge distances can be obtained by the Amap path planning API. Therefore, each edge in the edge set \(\mathbf{E}\) of the expressway network graph \(\mathbf{\hat{G}}\) corresponds to a unique edge distance, forming the edge distance set \(D\). The edge distance set \(D\) can be used as a non-negative weight for the directed edges of graph \(\mathbf{\hat{G}}\), forming an expressway network distance-weighted graph \(\mathbf{\hat{G}_{D}}=\left.<G,\ E,\ D>\right.\). **Definition 4**. : _Edge flow is defined as the number of vehicles that pass through both nodes of a directed edge \(e_{k}=\left\langle g_{i},g_{j}\right\rangle\in\mathbf{E}\). It is denoted as \(f(e_{k})\)._  
Similarly, the edge flow set F can be used as a non-negative weight for the directed edges of graph \(G\), forming an expressway network flow-weighted graph \(\mathbf{\hat{G}_{F}}=\left.<G,\ E,\ F>\right.\). It is worth noting that edge flow directly reflects the importance of directed edges: the larger the edge flow, the greater the traffic load on that road segment, indicating a higher level of importance in the entire road network. **Definition 5**.