## 4 Experiments and Results
### Validation Experiments
#### 4.2.1 Comparison of Expert Strategies  
In order to verify the effectiveness of the centralized APF strategy proposed in this paper for improving the swarm robustness, another APF strategy [46] based on dynamic topology relations is selected for comparative experiments. By defining potential truncation surfaces where the potential gradient becomes zero, the computational load of this strategy is significantly reduced, which makes it directly applicable to large-scale flocking control. However, based on the dynamic topology relations, this strategy can only achieve minimal potential energy of individuals in the local range [46], which indicates the unbalanced distribution of graph importance.  
In the following, we design a flocking control experiment of seven robots to compare the performance of the two expert strategies. We consider a swarm of \(N=7\) robots with a maximum communication radius of \(d_{max}=1.2\) m, a bandwidth-constraints of \(c=4\), and the sampling time of \(T_{s}=0.01\) s. The maximum velocity of the robot is set to be \(3\) m/s. The experiment is executed based on the same random initial state of the swarm, where the initial topology is a connected graph. Figure 3 shows the morphological changes of the swarm under the control of the two strategies.  
Figure 4 shows the variation of the single-step consistency evaluation metrics under the control of the two expert strategies in a specific episode.  
Figure 3: Variation of swarm morphology under the control of two expert strategies. (**a**) Three random initial forms of swarms. (**b**) The stable forms under the control of the APF strategy proposed in this paper. (**c**) The stable forms under the control of the APF strategy proposed in [46].  
It is worth noting that during the execution of the centralized expert strategy, information interaction exists between all nodes and the graph is fully connected. In order to ensure the objectivity and consistency of the GIE evaluation, the maximum communication distance is still assumed to exist. As long as the relative distance exceeds the maximum communication distance, it is considered that no edge exists between the two nodes when performing the entropy evaluation.  
Through the comparison, it can be found that although the APF strategy proposed achieves the velocity and one-hop distance consistency slightly slower than the APF strategy in [46], the stable values are almost the same. In terms of graph importance entropy, the APF strategy proposed in this paper significantly achieves a better performance, which proves that it can effectively achieve a balanced distribution of graph importance. The balanced distribution can also be intuitively reflected by the final stable form of the swarm. Since the APF strategy proposed in [46] is locally optimal, the final swarm morphology is influenced more by the initial morphology of the swarm, which is contrary to the goal of this paper to achieve a balanced distribution of graph importance and further improve the swarm robustness.  
Table 1 lists the values of the consistency evaluation metrics obtained by executing the two expert strategies separately on the test set. We can find that although the strategy proposed in this paper has smaller VR and DR than the strategy proposed in [46], it performs significantly better than the latter in terms of GIE, which validates its better performance in achieving the balanced distribution of graph importance. Furthermore, in terms of the SR, the expert strategy proposed in this paper is also more consistent with the goal of this paper to improve the swarm robustness. In a word, the expert APF strategy proposed in this paper realizes a balanced distribution of graph importance through more individual movements to improve the swarm robustness of the steady-state, and the velocity consistency and one-hop distance consistency before reaching the steady-state are partially sacrificed.