# An Improved Vicsek Model of Swarms Based on a New Neighbor Strategy Considering View and Distance
## 5 Results and Analysis
### Simulation and Analysis of SPVM  
In SPVM, view angle \(\omega\) and selection proportion \(p\) are the key parameters that determine the performance of the model. Therefore, this section will analyze the coupling effect of \(\omega\) and \(p\) on the performance of the model and the coupling relationship between \(\omega\) and \(p\). * Simulation of SPVM under noiseless conditions  
The time step \(k\)-order parameter \(V_{a}\) relationship of various Vicsek models, including VM, RVFVM, RNSVM, and SPVM, are compared in noiseless conditions. It is noticed that the original VM is a special case of SPVM with \(\omega=2\pi\) and \(p=1\); RVFVM is a special case of SPVM with \(p=1\); RNSVM is a special case of SPVM with \(\omega=2\pi\). In RVFVM, there is an optimal view angle \(\omega_{opt}\) which leads to the shortest convergence time \(T\) and \(\omega=\omega_{opt}=1.4\pi\) is adopted in the simulation. Similarly, regarding RNSVM, \(p=p_{opt}=0.4\) is adopted to perform the simulation. Regarding SPVM, the view angle and the selection proportion are set as \(\omega=1.2\pi\) and \(p=0.4\). The simulation results are shown in Figure 4a. It can be seen that the proposed SPVM reaches the ordered state (\(V_{a}=1\)) with the least time steps. It is worth mentioning that the value of view angle \(\omega\) set for RVFVM and the value of selection proportion \(p\) set for RNSVM are optimum. Additionally, SPVM can accelerate the convergence speed further by combining view restriction and remote neighbor strategy. Moreover, as shown in Figure 4b, SPVM also has the least number of neighbors. To sum up, the proposed SPVM can further improve the model's convergence performance while reducing neighbor information obtained from the environment simultaneously. The combinatory effect of view angle \(\omega\) and selection proportion \(p\) on the convergence time \(T\) of SPVM is shown in Figures 5 and 6. In the following simulations based on SPVM, \(\omega\) or \(p\) changes, while other parameters are set as: \(N=400,L=10,v_{0}=0.1\), \(R=1,\eta=0\). Firstly, the convergence time \(T\) as a function of \(\omega\) for different values of \(p\) is shown in Figure 5a. A smaller value of \(T\) means that the system reaches the ordered state faster, indicating a better convergence performance. From Figure 5a, a trend shows that for a fixed \(p\), when \(\omega\) increases, \(T\) decreases first, then increases. It means that for different cases of \(p=p_{0}\), there exists an optimal value of \(\omega=\omega_{opt,p=p_{0}}\) that leads to the shortest convergence time. When \(p\) is too small, convergence time \(T\) will get longer, but the relationship between \(\omega\) and \(T\) observed before will be lost. Then, \(\omega\) is set as a certain value and the relationship between \(p\) and \(T\) is shown in Figure 5b. For a fixed \(\omega\), when \(p\) increases, \(T\) decreases first and then increases, which means there exists an optimal value of \(p_{opt,\omega=\omega_{0}}\) that leads to the shortest convergence time when \(\omega=\omega_{0}\). Figure 6 demonstrates the relationship between view angle \(\omega\), selection proportion \(p\), and convergence time \(T\). Obviously, when \(\omega\) or \(p\) is too small, the convergence time \(T\) will be longer because particles in the system  
Figure 4: Comparisons of various Vicsek models. (**a**) \(V_{a}\) and (**b**) \(n\) as a function of time step \(k\). For other parameters: \(N=400,L=10,v_{0}=0.1,R=1,\eta=0\). Each data point is obtained by averaging over 300 different realizations. \(p\) or \(\omega\) both lead to fewer neighbors. Figure 8 demonstrates the relationship between \(\omega\), \(p\) and \(V_{stable}\). \(V_{stable}\) reaches its maximum when \(\omega=0.9\pi\) and \(p=0.6\), so by limiting the environment information particles obtain, the model can reach an ordered state of a higher level.