# Extraction of Catastrophe Boundary and Evolution of Expressway Traffic Flow State
## 3 Catastrophe Boundary Extraction with Spectral Clustering Analysis Algorithm
### Analysis of Traffic Characteristics  
The extraction results of the mutation boundary were analyzed, as shown in Figure 3, where green represents the free flow state, red represents the congested flow state, and blue represents the synchronized flow state. It can be seen from the figure that, compared with the free flow state and the congested flow state, the data points of the synchronized flow state were sparser, in line with the catastrophe characteristics of the cusp catastrophe theory. Its catastrophe boundary was \([-22.6,\,0]\). Then, the speed boundary before coordinate transformation was \([41.1\,\,\mathrm{km/h},\,63.7\,\,\mathrm{km/h}]\).  
In order to explore the versatility and universality of the method, several experiments were carried out by changing the parameters of the traffic simulation scenario to obtain the catastrophe boundary of the traffic flow state in different scenarios. The experimental scheme is shown in Table 3.  
The spectral clustering analysis algorithm was used to extract the speed catastrophe boundary from the data obtained from each simulation experiment, and the relationships between the catastrophe boundary and the weaving volume ratio, the length of weaving section, and the number of lanes were studied.  
As shown in Figure 4a, the catastrophe boundary of speed decreased with the increase in weaving volume ratio. When the weaving volume ratio was increased, the vehicles changed lanes more frequently in the weaving section, and the critical speed value of the traffic flow state was lower. With the increase in the length of the weaving section, as shown in Figure 4b, the speed catastrophe boundary showed an upward trend. At this time,  
Figure 3: Extraction results of catastrophe boundary.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Experimental Scheme** & **Experimental Parameters** & **Initial Value** & **Final Value** & **Increment** & **Remark** \\ \hline
1 & Weaving volume ratio & 0.05 & 0.35 & 0.1 & Each group of experiments was \\
2 & Length of weaving section (m) & 150 & 750 & 150 & only carried out according to the \\
3 & Number of lanes & 2 & 4 & 1 & other parameters remained \\  & & & & & unchanged. \\ \hline \hline \end{tabular}
\end{table}
Table 3: Summary of experimental scheme.  
vehics had more lane-changing opportunities during operation, and the traffic flow state had a higher critical speed value. As shown in Figure 4c, when the number of expressway lanes increased, vehicles could change from the outer lane to the inner lane, such that the outer lane had more lane-changing gaps, which improved the speed of traffic flow.  
In order to further analyze the sensitivity of various influencing factors to the catastrophe boundary of speed, the change rates of all experimental parameters were drawn as a box plot.  
It can be seen in Figure 5 that the weaving volume ratio was more sensitive to the speed catastrophe boundary. The change in weaving volume ratio can affect the traffic efficiency of the mainline and lead to significant changes in the traffic flow state of the weaving section. The length of the weaving section and the number of mainline lanes were less sensitive to the catastrophe boundary of speed. The main reason is that, with the  
Figure 4: Speed catastrophe boundary in different scenarios. (**a**) Weaving volume ratio—catastrophe boundary; (**b**) Length of weaving section—catastrophe boundary; (**c**) Number of lanes—catastrophe boundary.  
increase in weaving space, the interaction between lanes in the weaving section decreased gradually, and the traffic flow efficiency was higher. The critical speed changed slightly with a slight increase in input traffic volume.