## 4 Experiments and Results
### Distributed Control Strategy Comparison Experiments
#### 4.3.1 Test on Small-Scale Swarms  
Table 4 presents the evaluation results for the three strategies mentioned above on the test set. To provide a benchmark for comparison, the test results of the centralized APF strategy proposed in this paper is also listed in the table. The test results show that, the IDGCNs-based strategy does achieve the performance close to the expert template while respecting the communication constraints. Table 4 also shows that the control results of the two learning-based strategies are much better than the traditional distributed APF strategy in terms of SR. Based on the same initial states, the distributed APF strategy nearly completely fails to achieve the essential task of maintaining connectivity and avoiding splitting. Comparing the consistency evaluation metrics of the two learning-based control strategies, we can find that the metrics implemented based on IDGCNs are mostly better than those of the learning strategy based on DAGNNs, especially in the metric GIE and SR. The DAGNNs-based strategy is also generated by imitation learning with graph convolutional networks. However, it aims to achieve only velocity consistency, so the method performs poorly in terms of the swarm robustness. Figure 6 illustrates the final stable form reached by three different distributed control strategies for the same initial states in a specific episode. In addition, Figure 7 records the variation of the single-step consistency evaluation metrics of the two learning-based strategies during the test episode. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Strategy** & **VR** & **DR** & **GIE** & **TS** & **SR** \\ \hline Centralized APF strategy & \(\mathbf{-52.878}\) & \(\mathbf{-72.027}\) & \(\mathbf{1.891}\) & \(7.681\) & \(\mathbf{1.0}\) \\ IDGCNs-based strategy & \(\mathbf{-112.724}\) & \(\mathbf{-139.456}\) & \(1.881\) & \(11.507\) & \(0.945\) \\ DAGNNs-based strategy & \(\mathbf{-225.831}\) & \(\mathbf{-49.448}\) & \(1.669\) & \(16.223\) & \(0.755\) \\ Distributed APF strategy & \(\mathbf{-1199.25}\) & \(\mathbf{-21.055}\) & \(\mathbf{1.215}\) & \(\mathbf{4.293}\) & \(0.08\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Evaluation results under the control of the centralized expert strategy and the three distributed strategies. From the changes of swarm forms in Figures 3 and 6, it can be found that the learning strategy based on IDGCNs has learned from the expert strategy to achieve the balanced distribution of graph importance. At the same time, the stable form under the control of the DAGNNs-based strategy is fragile, and the swarm is completely split under the control of distributed APF strategy. From the variation of the single-step consistency evaluation metrics of the selected episode, it can be found that the time consumed to achieve consistency under the control of the two strategies is almost the same, and the IDGCNs-based strategy eventually achieves better velocity consistency and the DAGNNs-based strategy achieves better one-hop distance consistency. However, the difference of the final steady-state values is minimal. From the variation of the single-step GIE and the average GIE of all the test episodes, it can be found that the IDGCNs-based strategy is much better than the DAGNNs-based strategy. Thus, we believe that achieving the form with a balanced distribution of graph importance is the determining factor for the higher SR of IDGCNs-based control strategy to avoid swarm splitting. Figure 6: Variation of swarm morphology under the control of different control strategies in a specific episode. (**a**) The initial form of the swarm. (**b**) The stable form under the control of the centralized APF strategy. (**c**) The stable form under the control of the IDGCNs-based strategy. (**d**) The stable form under the control of the DAGNNs-based strategy. (**e**) The stable form under the control of the distributed APF strategy.