## 3 Route Planning Scheme Based on Scatter-GNN
### Problem Definition  
Since the number of high-speed railway lines in eastern China far exceeds that in western China, this paper defines this railway network as an edge graph structure network. We propose a method for solving station prediction in edge graph networks. The edge graph refers to the existence of some edge nodes in the graph. The specific definition is as follows: Given a railway network G, the number of stations \(V\) in G is \(n\), the connecting edges between two stations are called routes, and \(E\) is the number of routes. Then when G is neither fully connected nor empty, there are:  
\[0<E<\frac{n(n-1)}{2} \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Order** & **Route** \\ \hline
1 & Beijing–Tianjin–Jinan-Xuzhou–Bengbu–Nanjing–Shanghai \\
2 & Beijing–Shijiazhuang–Zhengzhou–Wuhan–Changsha–Guangzhou–Shenzhen–Jiulong \\
3 & Beijing–Chengde–Chaoyang–Fuxin–Shenyang–Tieling-Siping–Changchun–Harbin \\
4 & Shenyang–Anshan–Yingkou–Dalian \\
5 & Hangzhou–Ningbo–Taizhou–Wenzhou–Yuzhou–Xäamen–Shenzhen \\
6 & Nanjing–Hefei–Wuhan–Chongqing–Chengdu \\
7 & Lianyungang–Xuzhou–Shangqi–Zhengzhou–Luoyang–Xi’an–Baoji–Lanzhou–Xining–Wulumqi \\
8 & Shanghai–Hangzhou–Nanchang–Changsha–Guiyang–Kunming \\
9 & Qingdao–Jinan–Dezhou–Shijiazhuang–Taiyuan \\ \(\dots\) & \(\dots\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Example of datasets. where a possible value of \(E\) is \(nf(n)\), then the edge site of network G can be determined as:  
\[V=F(e) \tag{2}\]  
where \(e\) represents the number of neighbors around any site, and \(F\) is a logistic function that represents the true state of the result when \(e\) satisfies the following conditions:  
\[F(e)= \begin{cases}1,e\leq f(n)\\ 0,e>f(n)\end{cases}  \tag{3}\]  
the \(f\) in the above formula is an adaptive function, denoted as the mapping of \(f\) to \(F\):  
\[f\to F \tag{4}\]  
when \(n\) tends to infinity, if the value of \(V\) is only related to \(n\), and \(F^{\prime}\) indicates that the node degree is less than the independent variable, the edge graph node is finally defined as:  
\[V=F^{\prime}(f(n)) \tag{5}\]  
In Section 3.3, this paper focuses on deriving the solution process of the adaptive function. The direction of some high-speed railway lines and main stations can be described as shown in Figure 1 below. Among them, the black dotted line connects the possible passing stations of the Yinchuan-Chongqing high-speed railway. The black solid line represents the line segment that this paper focuses on, and the gray lines are other lines that actually exist. Figure 1: Schematic diagram of the forecast of the Yinchuan–Chongqing high-speed railway (HSR), station. from the pinch theorem we obtain:  
\[\lim_{n\rightarrow\infty}\frac{n(1+n)\sqrt{n}}{2\sqrt{\left(n^{2}+\frac{n}{2} \right)}}=\lim_{n\rightarrow\infty}\frac{n\sqrt{n}\sum_{n=1}^{n}\frac{n}{\frac{n ^{2}+n}{\sqrt{n^{2}+n}}}}{\sum_{n=1}^{n}\frac{1}{\sqrt{n^{2}+n}}}=\lim_{n \rightarrow\infty}nf(n)=\frac{n\sqrt{n}}{2} \tag{12}\]  
that is when \(n\rightarrow\infty\):  
\[f(n)=\frac{\sqrt{n}}{2} \tag{13}\]  
The above formula shows that when the number of stations in the network is large enough, the sum of the number of all routes tend to \(\frac{n\sqrt{n}}{2}\). Then site \(V\) in the network is defined as an edge site as:  
\[V=F^{\prime}\left(\frac{\sqrt{n}}{2}\right) \tag{14}\]  
Substitute 137 stations in the high-speed railway line into this function to calculate: \(f(137)=\frac{\sqrt{137}}{2}<6\). Then the edge site is recorded as a site whose number of adjacent sites is less than 6. The process of location search and potential neighbor feature representation is described as the Algorithm 1:  
```
0:



corresponding edge site neighbors. Dashed lines indicate whether there is a route between the two stations that are predicted. If it exists, a link is added between them and the corresponding black link is deleted, otherwise its potential neighbors are deleted.