## 4 Discussion  
According to the above analysis, we already know that the UAV system will converge to a flock finally, on condition that the initial interaction network is connected. Then, a more important question is coming, that is, how to accelerate the emergency of a flock. In the following paragraphs, the energy and time costs, as well as scalability effects of emerging a flock are evaluated from the point of statistical significance and strength quantitatively. To guarantee that all of the UAVs will converge to cohesive flocking finally, one condition must be satisfied, that is, the initial interaction topology of the UAVs should be connected. As the nearest-neighbor interaction rules have been adopted and every UAV has the same communication radius, the common communication radius becomes the only factor that can change the network connectivity. The value of the communication radius decides the scale of the interaction network and also the neighbor number of each UAV. The initial state of the UAV system is supposed to be given. Let \(D_{min}\) denote the minimal value of the communication radius that forces the interaction network to be connected. For these connected communication radii \(D\in[D_{min},\infty)\), a smaller value means fewer neighbors for each UAV. This situation further causes the decrease of computational complexity and calculated quantity of the whole system, as well as the reduction of energy consumption. The distance between any two UAVs is one of the proper variables to draw the convergence speed of UAV flocking and also reflects the stability of the UAV system. If the distance between any two UAVs becomes stable, then the system becomes stable. Therefore, energy measurements can be evaluated simply by the following order parameter  
\[EE(t)=\frac{\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}d_{ij}(t)}{N(N-1)/2}. \tag{22}\]  
where \(d_{ij}(t)=\parallel p_{i}(t)-p_{j}(t)\parallel,i,j\in\
u\). To assess the energy and time costs of a flock, numerical simulations of 80 UAVs are carried out under a series of connected communication radii. There are mainly three steps: First, the initial states of 80 UAVs are produced randomly. Second, connected communication radius' sample values are linearly selected from the interval \([2.8,8]\). Finally, starting from the same initial states of 80 UAVs, the experiment is carried out under each sample communication radius. Figure 3b shows the specific potential functions under these sample communication radii, and the simulation results are shown in Figure 3a. Different colors describe the simulation results under different sample communication radii. It is clear to see that the system converges to a flock faster under a smaller communication radius, although in which a longer journey (equaling to the initial individual distance minus the stable individual distance) should be covered to reach the stable individual distance \(d^{*}=\sqrt{D}\). Moreover, the above conclusion is obtained according to a fixed initial state and fixed group size. In order to make the conclusion more rigorous and more precise, each experiment is repeated over 100 trials that start with different initial states. Convergence time \(t^{c}\) is adopted as the order parameter to evaluate the time cost of each experiment. The definition of convergence time \(t^{c}\) is the minimal value of \(t_{k}\), which satisfies the following constraint condition,  
\[\frac{EE(t_{k})-EE(T)}{EE(T)}\leq K, \tag{23}\]  
where \(K=0.01\). The standard error bar is defined as  
\[\sigma=\frac{1}{\sqrt{n}}t_{c} \tag{24}\]  
where \(n=100\) is the experiment number. The relationships between convergence time and sample communication radius under different group sizes are shown in Figure 4. In all of these figures, the system converges to a flock within simulation time. For each given group size, the convergence time increases with the enlargement of communication radius most of the time when \(D\geq D_{min}\).