## 1 Introduction
### Related Work
Thus, the destruction of robots is often ignored. However, when the control strategy is migrated to physical robots, the swarm robustness must be considered, since the splitting of any individual is dangerous and can easily cause the collapse of the whole swarm. Recently, some researchers raised a topic from an adversarial perspective named private flocking [26; 27; 28], which is a similar topic to the swarm robustness. Private flocking emphasizes how individuals in a swarm can control themselves to hide the critical nodes and links, which can help avoid swarm splits due to the destruction of core individuals in adversarial environments. However, the lack of an effective quantitative evaluation metric for the performance of different strategies causes its conclusions to be less convincing and the optimization process to be slower. Besides, some of the few existing swarm robustness evaluation metrics require complex calculations [29] or excessive information [30] and are therefore less practical. Current flocking control strategies can be divided into two categories: centralized control and distributed control. Centralized control strategies plan and control robots' behaviors based on global observations. Taking all robot states into account, they can usually get a globally optimal and complete solution. Typical work of centralized control strategies includes a swarm intelligence (SI)-based centralized flocking solution proposed in [31] and a collision avoidance framework for robot swarms presented in [32]. The former is applied to optimize the wireless sensor network localization and network coverage [31], and the latter can navigate a swarm of autonomous nano-drones to complete the planting task [32]. However, the centralized control strategy always relies too much on the central node and result in a fragile swarm. The larger the swarm scale is, the severer the problem becomes. In contrast, distributed control strategies are built based on local observations and communications. All robots are of equal importance and make their decisions independently, resulting in a uniform computational burden across the robot swarm. [PERSON] et al. proposed a dynamic group learning distributed swarm optimization method, which could be extended for the large-scale cloud workflow scheduling [33]. In [34], [PERSON] et al. presented a distributed swarm control framework for path following, where the dynamic models and velocity measures of underactuated unmanned marine vehicles are inaccurate and uncertain. However, because of the limited observation range and communication range, distributed control strategies easily fall into a local optimum. Therefore, there is a large gap of performance between distributed control strategies and centralized control strategies, which encourages researchers to improve distributed control strategies. In order to combine the advantages of centralized control strategies and distributed control strategies, some researchers have implemented imitation learning for flocking control in recent years [35; 36]. Based on the distributed structure and local observations, behavior cloning of centralized expert strategies is performed, aiming to obtain distributed control strategies of which control effects are close to the global optimums. As a learning-based method, imitation learning also faces the problem of state representation. Currently, the most popular solution is to combine imitation learning with graph neural networks (GNNs) [37; 38; 39; 40; 41]. GNNs can effectively extract and uniformly describe the feature of non-Euclidean data and can further help generate scalable distributed strategies. Until now, the GNNs-based imitation learning methods have been successfully applied in multi-robot path planning [37], multi-robot formation control [38], and multi-robot task assignment [39]. Various graph-based networks such as graph recurrent neural networks [41] and graph attention networks [39] are proposed and utilized in recent years.