# An Improved Vicsek Model of Swarms Based on a New Neighbor Strategy Considering View and Distance
## 5 Results and Analysis
### Simulation and Analysis of DPVM  
As is shown in Figure (a)a, SPVM cannot reach the balance between the faster convergence speed and the higher stabilized consensus level in noisy conditions. In DPVM, the updating functions of view angle \(\omega\) and selection proportion \(p\) are the key factors determining the model's performance. Therefore, simulations are conducted to verify that if the updating functions of \(\omega\) and \(p\) are properly designed, DPVM will be able to reach the balance.  
Since SPVM(1) can reach and oscillate around its maximum consensus level in less than 500 steps, SPVM(1) represents the model of a fast convergence speed. SPVM(4) has the highest value of \(V_{stable}\), and thus, SPVM(4) is the model that has the highest stabilized consensus level among all. Therefore, SPVM(1) and SPVM(4) are used to compare with DPVM from aspects of convergence speed and the stabilized consensus level. Simulation based on SPVM(1), SPVM(4), and DPVM is shown in Figure (b)b. In order to achieve a better performance, we have compared several function designs of the updating functions of view angle \(\omega\) and selection proportion \(p\). Eventually, we set the functions as Equations (9) and (10). In the simulation, DPVM demonstrates a balance between faster convergence speed and higher stabilized consensus level.  
\[\omega_{i}(k+1)=\left\{\begin{matrix}2\pi&0\leq V_{local}(k)<0.72\\ 0.9\pi&0.72\leq V_{local}(k)\leq 1\end{matrix}\right. \tag{9}\]  
\[p_{i}(k+1)=\left\{\begin{matrix}0.7&0\leq V_{local}(k)<0.72\\ 0.6&0.72\leq V_{local}(k)\leq 1\end{matrix}\right. \tag{10}\]  
The simulation result has been demonstrated in Table 2. Quantitatively, \(V_{stable}\) of SPVM(1) equals to 0.7/83; \(V_{stable}\) of SPVM(4) equals to 0.8032; \(V_{stable}\) of DPVM equals to 0.7973. In order to evaluate the convergence speed of the models, the threshold value of the definition of convergence time \(T\) need to be adjusted. Because of the presence of noise, the maximum value \(V_{a}\) can reach is 0.8 based on current parameters, so the convergence speed cannot be measured if the threshold of \(T\) remains at 0.99. Therefore, the threshold value of the definition of convergence time \(T\) is adjusted from 0.99 to 80% of the maximum value \(V_{a}\) can reach under the present parameters. Convergence time \(T\) here is, thus, redefined as the time steps required by the model for \(V_{a}\) to surpass 0.64 from the random initial state when \(V_{a}\) is close to 0. The value of 0.64 can effectively serve as a threshold value because separate groups of particles moving in an orderly manner have been formed when \(V_{a}=0.64\) according to our observation. \(T_{threshold}=0.64\) of SPVM(1) equals to 55; \(T_{threshold}=0.64\) of SPVM(4) equals to 1579; \(T_{threshold}=0.64\) of DPVM equals to 96. By comparing these two performance indicators of the three models, it can be concluded that DPVM is the only  
Figure 9: Order parameter \(V_{a}\) as a function of time step \(k\). Each data point is obtained by averaging over 20 (**a**) and 30 (**b**) random initializations. \(N=400,L=10,v_{0}=0.1,R=1,\eta=1\).  
model that is capable of reaching a balance between faster convergence speed and higher stabilized consensus level.