# Exploration of Swarm Dynamics Emerging from Asymmetry
## 6 Discussion  
### Overview of Results  
According to their dynamical motions, we have roughly classified the patterns including already found ones (S1 to S17) shown in Figure 2 and uninteresting ones (S0a and S0b), into several categories. This classification is shown in Table 3. The results from the seven fitness functions proposed in this paper are also in this table.  
Defining a good measure was found to be not easy: there are not few cases where even uninteresting patterns could get a high fitness score. A swarm tends to result in a segregating pattern, which is frequently observed but uninteresting, when one or more types have a too small perception range and a too large speed. Therefore, introducing some terms of limiting into the fitness functions may be effective to prevent an easy breaking of the structure.  
The recipes gained through the optimization are shown in Figure 8. We can see from this table that  
1. Some types have a small perception range \(R\) and a large normal speed \(V_{n}\) (e.g., Type 1 in FF-A, Type 1 in FF-B, Type 2 in FF-C, Types 2 and 3 in FF-F and Type 3 in FF-G). On the contrary, some other types have a large \(R\) and a small \(V_{n}\) (e.g., Type 1 in FF-C, Types 1 and 3 in FF-D, Type 1 in FF-F, Types 1 and 2 in FF-G).
2. Similarly, some types have a small cohesion coefficient \(c_{1}\) and a large separation coefficient \(c_{3}\) (e.g., Types 1 and 2 in FF-C), and vice versa.
3. In FF-E, Types 1 and 3 share the almost identical KPs.  
Figure 12: Representative dynamics of the swarm obtained from FF-F. (**a**) The snapshot at a timestep \(t\); (**b**) at \(t+10\); (**c**) at \(t+15\); (**d**) at \(t+35\); (**e**) at \(t+40\); (**f**) at \(t+60\); (**g**) at \(t+75\); and (**h**) at \(t+105\). The shape continues to deform from an egg-like structure to eventually fission into several aggregations.  
Especially, as can be seen in items 1 and 2 above, the relationships between the perception range \(R\) and the normal speed \(V_{n}\) as well as between the cohesion coefficient \(c_{1}\) and the separation coefficient \(c_{3}\) seem important to control the dispersion behaviors.  
From the above results together with the dynamics obtained, we discuss the relationship between the recipe (KPs) and the resultant swarm patterns. Dispersive motions like FFs-A, B, C and G are observed when: the swarm has one or more types whose perception range \(R\) is small whereas normal speed \(V_{n}\) is large; and/or separation coefficient \(c_{3}\) is large whereas cohesion coefficient \(c_{1}\) is small. Therefore, it seems better to keep \(c_{1}\) large and \(c_{3}\) small, as well as to keep \(R\) large and \(V_{n}\) small, in order to create a linked formation preventing from a dispersion.  
Figure 8 also summarizes the statistical tendency, telling us that:  
1. As the standard deviations are small and the mean values are relatively high (approximately 0.8) in \(c_{2}\) (aligning coefficient), \(c_{4}\) (randomness) and \(c_{5}\) (tendency of self-propulsion), implying that these parameters are almost common and less sensitive to the definition of fitness functions than the other ones.
2. On the other hand, diversity is large for \(R\) (perception range), \(V_{n}\) (normal speed), \(V_{m}\) (maximum speed), \(c_{1}\) (cohesive coefficient) and \(c_{3}\) (separating coefficient), implying that these parameters are sensitive to the definition of fitness functions.
3. In addition, the mean values of \(c_{1}\) tends to be large and that of \(c_{3}\) is small.