# The Influences of Self-Introspection and Credit Evaluation on Self-Organized Flocking
## 3 Simulations and Discussions
### Noise-Free Calculation and Analysis
#### 3.1.1 Influence of Parameters on the Convergence Time \(T\)
Here, 300 individuals move into the square 2D area (\(L=5\)). The speed \(v\) of individuals increases from 0.03 to 1.03, and the interaction radius \(r\) increases from 0.4 to 1.4. The convergence time was obtained from the average of 500 independent noise-free runs. Figure 3: Convergence times \(T\) under different individual numbers \(N\) and interaction radii \(r\) for (**a**) SIM (\(\alpha=1\)) and (**b**) CEM (\(\beta=1\)). \(N\) individuals move into the square 2D area (\(L=5\)). The number of individuals \(N\) increases from 50 to 550, and the interaction radii \(r\) increases from 0.5 to 1.5. The convergence time \(T\) is obtained from the average of 500 independent noise-free runs. became worse, especially when \(\alpha>3\) and \(\beta>3\). Meanwhile, we also verified the above conclusion from Figure 5 again that the convergence time will decrease along with the increase in the number of individuals \(N\) and their interaction radius \(r\). It can also be seen from Figure 5, that within a certain range, if the flocking parameters were fixed, then the convergence time decreased with the increase in model parameters \(\alpha\) or \(\beta\). However, the fitting slope shows different trends. The fitting slope in Figure 5 increases with the increase in \(\alpha\), while the variation shown in Figure 5 is exactly the opposite, and the fitting slope decreases with the increase in \(\beta\). This is because the increase in \(\alpha\) weakened the effect of local consistency in the SIM, gradually tending toward a change in orientation with a fixed weight, as well as weakening the influence of parameters \(N\) and \(r\) on the convergence time. In sharp contrast to the SIM, as the model parameter \(\beta\) of the CEM increased, the effect of local consistency would not be weakened due to the introduction of weight normalization in Equation (7), and thus the fitted slope merely decreased slightly. To show the relationship between \(m\) and \(\alpha\) and \(\beta\) more clearly, further numerical simulations were performed, and the results are shown in Figure 6. Based on the metric of the convergence time, there were linear relationships between the model parameters and \(m\) in Equation (10); that is, \(m=0.0714\alpha\), and \(m=-0.0417\beta\). Therefore, Equation (10) can be rewritten as Equations (11) and (12) to represent the relationships among the convergence time \(T\), flocking parameters (\(r\), \(N\)), and model parameters (\(\alpha\), \(\beta\)) of the SIM and CEM, respectively. \[T\sim\left(r^{2}\ln N\right)^{0.0714\alpha}, \tag{11}\]  
\[T\sim\left(r^{2}\ln N\right)^{-0.0417\beta}. \tag{12}\]  
However, as the individual's speed increased, these relationships were no longer valid. The relationships between \(m\) and \(\alpha\), \(\beta\) are nonlinear and irregular when \(v\) is large. In summary, the model parameters and flocking parameters affected the flocking behavior together in a certain range of parameters. Specifically, the connectivity of the network whose main parameters were the interaction radius and density as well as the interaction mechanisms of individuals jointly affected flocking. Figure 5: Convergence time \(T\) as a function of \(r^{2}\ln N\). The different parameters \(\alpha\) of the (**a**) SIM and \(\beta\) of the (**b**) CEM can be fitted well in the double logarithmic coordinate. Speed is fixed at \(v=0.03\), and control parameters \(r\) and \(N\) vary from 0.5 to 1.5 and from 50 to 550, respectively. Each point is obtained from the average of 500 independent noise-free runs.