# The Influences of Self-Introspection and Credit Evaluation on Self-Organized Flocking
## 3 Simulations and Discussions
### Noise-Free Calculation and Analysis
#### 3.1.2 Optimal Values of the Model Parameters  
Although the quantitative relationship among the parameters in the models was explored from the perspective of the convergence time, Figure 5 also shows that the convergence times for specific flocking parameters and model parameters did not always follow such a quantitative relationship. In order to find the optimal values of \(\alpha\) and \(\beta\), we carried out simulations with different values of \(\alpha\) and \(\beta\) in the range of \([0,5]\) using the SIM and CEM respectively. By comparing the convergence time \(T\) and cumulative global consistency \(S\), the optimal values of \(\alpha\) and \(\beta\), denoted by \(\alpha_{\text{opt}}\) and \(\beta_{\text{opt}}\), respectively, were determined under different flocking parameters. We first investigated the optimal values under the metric \(T\). Figure 7a shows that under the constant density and interaction radius of the group, the optimal model parameter \(\alpha_{\text{opt}}\) of the SIM gradually decreased as the individual speed \(v\) increased. In particular, the optimal parameter \(\alpha=0\) when the speed \(v\geq 1.2\), which is exactly the same as the VM. This means that the [PERSON] model has more advantages in terms of convergence time when the speed \(v\) is higher, while the model parameter \(\beta_{\text{opt}}\) of the CEM slightly decreased with the increase in \(v\). The interaction radius of an individual also affects the optimal value of the model, which can be drawn from Figure 7b. When the density and speed of the group were constant, the optimal model parameter \(\alpha\) of the SIM increased first and then decreased with the growth of the individual interaction radius \(r\). Conversely, the optimal parameter \(\beta\) of the CEM decreased first and then increased with the increase in the interaction radius \(r\). We tried to understand these two different trends from the perspective of the mechanisms of the two models. Under the condition of a small interaction radius, the interaction mechanism of self-introspection (SIM) made the individual have fewer neighbors to refer to, and the average orientation of the neighbors was not particularly reliable. Therefore, it is not necessary for the individual to use the interaction mechanism of self-introspection to move; instead, individuals adopt a fixed-weight interaction mechanism, which means \(\alpha\) is large. A larger interaction radius increased the number of neighbors, and reliable information about the neighbors made individuals interact with neighbors in a more introspective way (\(\alpha_{\text{opt}}\) became smaller). Especially when \(r\) increased to a certain extent (\(r>1.8\)), individuals were more inclined to the average orientation of the neighbors (\(\alpha=0\), [PERSON] model). For \(\beta_{\text{opt}}\) of the CEM, when \(r\) was small, there were fewer neighbors, and an individual could not give much credit to a neighbor even if he or she was more reliable, and thus \(\beta_{\text{opt}}\) was usually under two. In the case of larger \(r\) values, this means that the increase in neighbors would make individuals pay more attention to the neighbors with better alignment, and thus \(\beta_{\text{opt}}\) was large and usually around three. As for the density of the individuals, as shown in Figure 7c, if the speed \(v\) and the interaction radius \(r\) of flocking  
Figure 6: Exponent \(m\) as a function of model parameters \(\alpha\) and \(\beta\). The range of model parameters is from 0\(\sim\)5, and other parameters are the same as in Figure 5. Each point was obtained from the average of 500 independent noise-free runs. were fixed, then the increase in density would barely affect the optimal parameters of the SIM, which only fluctuated around 2.5\(\sim\)3.5. However, the optimal parameter \(\beta_{\text{opt}}\) of the CEM would increase first and then decrease slightly.