# 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 \(T\)  
The collective system can be seen as a complex social network, where the flocking parameters \(r\) and \(N\) under a fixed speed \(v\) directly affect the connectivity of the network. Otherwise, different speeds of individuals affect the frequency of information exchange in complex networks. Therefore, the flocking parameters (\(v,r,N\)) and model parameters (\(\alpha,\beta\)) jointly determine the interaction mechanisms of individuals. The impact of the model parameters and flocking parameters on the collective behavior were analyzed qualitatively and quantitatively with the convergence time \(T\) as the metric. We kept \(L\) fixed and changed \(N\) to change the density \(\rho\), where \(\rho=N/L^{2}\). The convergence time \(T\) has been described before, and we set a threshold of \(\phi_{three}=0.95\). We first kept the speed of flocking individuals fixed as a constant and analyzed the influence of the density in the system and the interaction radius \(r\) on the convergence time \(T\). It can be seen from Figure 3 that the convergence time of the two models decreased as the interaction radius and density increased, which is similar to the original [PERSON] model. The main reason for this is \(\rho\) and \(r\) are not only the main parameters affecting the local consistency \(\phi_{i}\) but also the main factors of the topology network of the group. If the flocking density \(\rho\) is kept as a constant, then increasing the interaction radius \(r\) within a certain range is equivalent to increasing the connectivity of the network, thus adding more references (neighbors) to each individual. Therefore, the time from disorder to order of the collective will be greatly reduced. On the other hand, if the interaction radius \(r\) is fixed, and the density \(\rho\) is increased, then the connectivity of the local network will be increased, the local information obtained by the individual will be more valuable, and the convergence time of the group will also be reduced. It is obvious that reducing the flocking density or the interaction radius will gradually lead to a less connected network, and thus the convergence time will increase. To sum up, the density and interaction radius are both the main flocking parameters that affect the complex network of a group. Next, the speed was selected as the main parameter and combined with the interaction radius to further analyze the impact of the flocking parameters on collective movement. Through the simulation analysis, we found that the amplitude of the individual speed also affected the convergence time of the flocking. Figure 4 shows a comparison of the convergence times \(T\) for the VM, SIM, and CEM under different flocking parameters (\(v\), \(r\)). Within a certain range, increasing the speed can significantly reduce the convergence time of flocking, as shown in Figure 4. This is because increasing the speed is equivalent to increasing the frequency of information exchange between individuals. Through the above analysis, it can be known that the flocking parameters \((v,r,N)\) determined the convergence time \(T\) of the group, given that the model parameter \(\alpha=\beta=1\). In order to further explore the quantitative relationship between the two parameters \(r\) and \(N\) that affected the connectivity of the topology network, the model parameters (\(\alpha\), \(\beta\)), and the convergence time \(T\), extensive numerical simulations were performed with a fixed velocity amplitude (\(v=0.03\)). The results are shown in Figure 5, and the relationship between them was found to be  
\[T\sim\left(r^{2}\ln N\right)^{m}, \tag{10}\]  
where \(m\) is the slope of the double logarithmic linear fitting, which is mainly determined by the model parameter \(\alpha\) or \(\beta\). However, as the model parameters increased, this linearity  
Figure 4: Convergence times under different speeds \(v\) and interaction radii \(r\) for three models. From top to bottom are the VM, CEM (\(\beta=1\)), and SIM (\(\alpha=1\)).