## 2 Problem Formulation  
### Description of the Robot Swarm  
We consider a distributed swarm consisting of \(N\) omnidirectional robots in the two-dimensional space. The sampling time interval is denoted as \(T_{s}\), and \(n\) denotes the sampling sequence number. At time \(nT_{s}\), the positions, velocities, action policies and observations of robots are denoted by \(\mathbf{R}_{n}=[\mathbf{r}_{1,n},\mathbf{r}_{2,n},\cdots,\mathbf{r}_{N,n}]^{ \mathsf{T}}\in\mathbb{R}^{N\times 2}\), \(\mathbf{V}_{n}=[\mathbf{v}_{1,n},\mathbf{v}_{2,n},\cdots,\mathbf{v}_{N,n}]^{ \mathsf{T}}\in\mathbb{R}^{N\times 2}\), \(\mathbf{U}_{n}=[\mathbf{u}_{1,n},\mathbf{u}_{2,n},\cdots,\mathbf{u}_{N,n}]^{ \mathsf{T}}\in\mathbb{R}^{N\times 2}\) and \(\mathbf{X}_{n}=[\mathbf{x}_{1,n},\mathbf{x}_{2,n},\cdots,\mathbf{x}_{N,n}]^{ \mathsf{T}}\in\mathbb{R}^{N\times F}\), respectively, where \(F\) represents the dimension of observation vector. In the following, if there is no superscript, the default is to represent the information at time \(nT_{s}\). Besides, since omnidirectional robots are mode led as discrete systems, \(\mathbf{U}_{n}\) will last a time interval, from the moment \(nT_{s}\) to the moment \((n+1)T_{s}\). The discrete dynamic model of the robot is described as:  
\[\left\{\begin{array}{l}\mathbf{R}_{n+1}-\mathbf{R}_{n}=\mathbf{V}_{n}T_{s}+ \frac{\mathbf{U}_{n}}{2}T_{s}^{2},\\ \mathbf{V}_{n+1}-\mathbf{V}_{n}=\mathbf{U}_{n}T_{s}.\end{array}\right. \tag{1}\]  
We assume the existence of time-varying local communication constraints, and further assume that each individual can only obtain information from at most the \(c\) nearest individuals in the communication area, where the number of \(c\) is related to the ability of the communication equipment. At this point, even if the distance between robots is less than the maximum communication distance, it is uncertain that the bidirectional interaction can be achieved, so the communication relationship between nodes can only be represented by a directed graph.  
In this work, the robot swarm is described by a directed graph \(\mathcal{G}^{d}=(\mathcal{V},\mathcal{E})\), of which the vertex set \(\mathcal{V}=(v_{1},v_{2},\cdots,v_{N})\) and the edge set \(\mathcal{E}\) represent the robot nodes and the communication links, respectively. The bidirectional neighbors of \(v_{i}\) are defined as \(\mathcal{N}(v_{i})=\{v_{j}|\|\mathbf{r}_{i}-\mathbf{r}_{j}\|<d_{max},i\
eq j\}\), where \(d_{max}\) is the maximum communication distance. However, since the communication bandwidth of \(v_{i}\) is restricted, the communication links between \(v_{i}\) and its bidirectional neighbors may be blocked. Therefore, we further define directed neighbors of \(v_{i}\) as \(\mathcal{N}^{d}(v_{i})=\{v_{j}|v_{j}\in\mathcal{N}(v_{i})\) and \(l_{ij}=1\}\), where \(l_{ij}=1\) indicates that \(v_{j}\) is one of the \(c\) nearest individuals in the communication area. We assume that \(v_{i}\) can only receive data from \(\mathcal{N}^{d}(v_{i})\), reference information from other individuals in the communication area will be ignored to ensure the communication quality and reduce the burden of memory space. The directed edge which starts from \(v_{j}\) and ends at \(v_{i}\) is denoted as \(e_{ji}^{d}=\langle v_{j},v_{i}\rangle\). Finally, the edge set is rewritten to \(\mathcal{E}=\{e_{ji}^{d}|\forall v_{i}\in\mathcal{V},v_{j}\in\mathcal{N}^{d}( v_{i})\}\).  
For the directed graphs, the number of edges pointing to \(v_{i}\) is called the degree of \(v_{i}\) and is denoted as \(\text{deg}(v_{i})\). A vertex whose observations can be received by the vertex \(v_{i}\) after at least \(k\) times data transmissions is called a \(k\)-hop neighbor of vertex \(v_{i}\), and the \(0\)-hop neighbor is the node itself. The subgraph containing all the \(\bar{k}\)-hop (\(\bar{k}=0,1,2,\cdots,k\)) neighbors and the edges between them are called the \(k\)-subgraph of vertex \(v_{i}\).