## 4 Experiments and Results
### Following Experiment  
In the process of flocking control, individuals refer to the behaviors of neighboring nodes to adjust their actions. As a result, in a swarm composed of homogeneous robots, without the introduction of external guidance, the final stable form and velocity depend primarily on the initial state of the swarm, which has a significant degree of uncertainty. In the following, we introduce the leader nodes to the swarm to guide the behaviors of neighboring robots, forming a hierarchical control structure of leader-follower swarm. The following task requires the flocking control strategy to achieve the following motion \begin{table}
\begin{tabular}{l l c c c c c} \hline \hline
**Scale** & **Strategy** & **VR** & **DR** & **GIE** & **TS** & **SR** \\ \hline \multirow{3}{*}{20} & IDGCNs-based strategy & \(-\)289.071 & \(-\)416.025 & 2.919 & 169.834 & 0.955 \\  & DAGNNs-based strategy & \(-\)1790.77 & \(-\)3524.47 & 2.911 & 289.037 & 0.75 \\  & Distributed APF strategy & \(-\)9853.81 & \(-\)41.796 & 2.374 & 7.491 & 0.0 \\ \hline \multirow{3}{*}{30} & IDGCNs-based strategy & \(-\)380.834 & \(-\)2858.52 & 3.302 & 154.626 & 0.97 \\  & DAGNNs-based strategy & \(-\)1411.9 & \(-\)3248.99 & 3.299 & 290.176 & 0.83 \\  & Distributed APF strategy & \(-\)11,386.1 & \(-\)40.631 & 2.777 & 7.499 & 0.0 \\ \hline \multirow{3}{*}{40} & IDGCNs-based strategy & \(-\)535.422 & \(-\)2902.76 & 3.571 & 144.961 & 0.955 \\  & DAGNNs-based strategy & \(-\)750.481 & \(-\)3224.67 & 3.573 & 287.891 & 0.885 \\ \cline{1-1}  & Distributed APF strategy & \(-\)12,382.8 & \(-\)43.578 & 3.043 & 8.971 & 0.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Evaluation results of different swarm scales under the control of the three distributed strategies. and trajectory imitation of the very few leaders with only local information. Since the leader's identity cannot be recognized by other individuals, its behavior has the same reference value as that of other neighboring nodes, and this paper refers such leader nodes as implicit leaders. We assume that the swarm consists of one leader and six followers. The initial form is a regular hexagonal where the leader is in the center, and the initial position and velocity of each individual are appended with Gaussian noise. The behavior of the implicit leader is preset to a sinusoidal motion with a maximum velocity of 0.8 m/s and a maximum acceleration of 0.5 m/s2. We verify whether the three distributed control strategies can effectively realize the following motion while avoiding swarm splitting and maintaining consistency in velocity, one-hop distance, and graph importance. The evaluation results are shown in Table 6.  
Since the initial form of the swarm exhibits symmetric and tight aggregation, the values of the consistency evaluation metrics are almost the same under the control of the IDGCNs-based and DAGNNs-based learning strategies. In terms of the success rate of avoiding swarm splitting, both of the learning strategies are significantly better than the distributed expert strategy. Figure 8 plots the sinusoidal motion trajectories and some intermediate morphology of the swarm under the control of the three strategies, where the implicit leader is marked by a larger red dot. It can be found from the trajectories that by executing the IDGCNs-based strategy, all the individuals achieve the imitation of the sinusoidal motion of the implicit leader, and the motion trajectory exhibits as an approximate standard sinusoidal curve. In the trajectories realized by the distributed APF strategy and the DAGNNs-based strategy, the nodes located along the direction vertical to the motion direction generate trajectories significantly deformed compared with the standard sinusoidal curve. In terms of the intermediate morphology of the swarm, the following motion implemented based on IDGCNs can maintain a more stable swarm morphology during the following process. Furthermore, under the control of the IDGCNs-based strategy, the links are more abundant, and the swarm is more resistant to interference.