## 4 Discussion  
This study examined the descriptive parameter related to the global state transition of group separation and recombination and the hysteresis concerning the global state transitions. In the candidate descriptive parameters analysis in each global state, including the stretched and divided state, the speed of the leading cyclist (V\({}_{leader}\)) was higher than in other states (Figure 1a). The speed of a leading cyclist may be too high for the follower to sustain when approaching maximal sustainable output when the group stretches or separates. According to [PERSON] et al. [11], group separation in a cycling race occurs when a drafting or following cyclist cannot sustain the pace of the leading rider even by drafting. Therefore, group stretch and separation are a consequence of increases in competition speeds that approach cyclists' maximal sustainable outputs. The time differences between the leader and the front of the main group (\(\Delta\)T) became larger following the state transition from dense to stretched, divided, and escape and dense state. This suggests that time difference could be regarded as an order parameter corresponding to the global state transition defined by [PERSON], [PERSON], and [PERSON] [2]. When focusing on the separation and recombination, there was a significant speed difference between the speed of the leading cyclist and the velocity of the front of the main group (\(\Delta\)V); that is, there was a significant difference between the case where the state was maintained and the case where the group was separated from a stretched to divided state, or the group was recombined from a divided to stretched state (Figure 2c,g). However, the speed of the leading cyclist did not significantly differ between the separation and recombination (Figure 2a,e). The time difference between the speed of the leading cyclist and the velocity of the front of the main group was considered to have a relative relationship between the speed of the leader and the main group. To examine the relationship between the speed of the leader and the main group, this study analyzed the speed of the leader and the speed difference between the leader and the main group when the group separated, the stretched group was maintained, the group rejoined, and the divided state was maintained. When the group was separated, the leader's velocity tended to be higher, and the speed difference was positive; that is, the leader tended to be faster than the main group compared to when the group remained stretched and was not divided. This suggests that for the group to be separated, it is necessary to make a difference in velocity and increase the velocity. When speeds are low relative to cyclists' maximal speed capabilities, it is challenging for cyclists to \"attack\" and escape from the group because chasing cyclists have abundant energy to chase the attacker and to reduce any separations generated by the attacking cyclist. Accelerating to 14 m/s or more and escaping from the group requires a further speed difference of 1.0 m/s or more; that is, to escape from a pack, an attacking cyclist needs to attain speeds at least 1.0 m/s more than the followers. However, our observation was of an elite-level men's velodrome race; hence, this may be applicable only to narrow circumstances. [PERSON] [13] introduced a random acceleration parameter (RAP) in the range of 0.0-2.0 m/s to the simulation of a cycling competition. RAP is a random acceleration added to the velocity of each agent, which was incorporated into a general algorithm based on a simulation of a flock of birds [11] to better reproduce a cycling competition. In [PERSON]'s simulation, RAP simulated cyclists' adjustments in speed to generate passing between cyclists at comparatively low speeds, but RAP was not high enough to cause separations in the peloton.