# The Influences of Self-Introspection and Credit Evaluation on Self-Organized Flocking
## 3 Simulations and Discussions
### Simulations with Noise  
All the previous simulation calculations were based on the noise-free case. The group would eventually achieve global consistency, and the value of the global consistency \(\phi\) would be close to one. However, groups are generally influenced by noise from individuals, which reflects the uncertainty and the errors of the individuals' motions. In order to show the robustness of the two proposed models, noise was taken into account for further analysis. For the purpose of testing the performances of the proposed models and comparing them to that of the Vicsek model, noise with the same amplitude, where \(\Delta\theta\in[-\eta/2,\eta/2]\), was added to Equations (2), (5), and (6). The noise was selected to be as strong as \(\eta\) = 0\(\sim\)4.5 to test the stability of these models. Using Equation (9), the final global consistency values \(\phi_{\eta}\) of the VM, SIM, and CEM are shown in Figure 10. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Simulation Settings**} & \multicolumn{3}{c}{**Convergence Time \(T\)**} \\ \hline
**Number of Individuals N** & **Models** & \(r=0.5\) & \(r=1\) & \(r=1.5\) \\ \hline \multirow{4}{*}{100} & VM & 139.08 & 18.80 & 5.42 \\  & SIM & **112.81** & **14.38** & 5.18 \\  & CEM & 135.22 & 15.60 & **4.35** \\ \hline \multirow{4}{*}{300} & VM & 93.82 & 14.16 & 4.74 \\  & SIM & **57.39** & **9.07** & 4.30 \\  & CEM & 86.12 & 10.78 & **3.75** \\ \hline \multirow{4}{*}{500} & VM & 84.93 & 12.93 & 4.70 \\  & SIM & **43.65** & **8.33** & 4.38 \\ \cline{1-1}  & CEM & 78.51 & 10.45 & **3.82** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of model convergence effects. In the noise-free case, it can be known from the above discussions that the flocking would eventually achieve a global consistency \(\phi=1\). However, after adding noise, the final global consistency would be less than one. Figure 10 shows the effects of different noise amplitudes on \(\phi_{\eta}\) with different densities for the three models (VM, SIM, and CEM). Here, we chose the flocking parameters (\(v=0.03\), \(r=1\)), and the model parameters here (\(\alpha,\beta\)) were selected from the results in Figure 8c. In the case of lower noise, the global consistency could be maintained between 0.9 and 1. As discussed above, the SIM has advantages in terms of convergence time in the noise-free case, but it can be seen from Figure 10 that the final global consistency \(\phi_{\eta}\) of the SIM with noise was somewhat poor. In particular, when the noise was larger (e.g., \(\eta=2\)\(\sim\)4.8), the ability of the SIM to resist noise was significantly weaker than the VM and CEM, regardless of the number of individuals. For a large number of individuals (Figure 10c,d), in the case with more noise, the performances of the VM and CEM against the noise were similar. As mentioned above, individuals in a group may have heterogeneous features. We now investigate the case where the individuals in the group behave differently. We assumed that the group included two kinds of individuals behaving according to the SIM and CEM, whose numbers were \(N_{I}\) and \(N-N_{I}\), respectively. We focused on the global consistencies and stabilities against noise for the groups with different \(N_{I}\) values under specific parameters. We set the flocking parameters to \(v=0.03\), \(r=1\), and \(N=300\), and the model parameters were \(\alpha=\beta=1\). The noise amplitude increased from 0 to 4.8 in the simulation. Numerical simulations were performed based on the above parameters, and the final global consistency \(\phi_{\eta}\) was recorded. The flocking behavior showed characteristics different from those of a single type of group under the influence of noise, as shown in Figure 11. Meanwhile, the groups with different proportions of behaviors showed different noise resistance abilities.