## 2 Problem Description
### Hierarchical Communication Topology  
The UAV swarm is divided into several distributed subsets. Communication exists within each subset and among subsets, hence hierarchical communication topology is established. In the application process, the swarm can cooperate to assign and complete tasks through the two-layer mechanism of inter-group cooperation and intra-group coordination based on hierarchical communication topology. The structure of the distributed subsets adopted integrates the advantages of both centralized structure and fully distributed structure to realize \"global centralization and local autonomy\", which avoids the problems of low redundancy and the heavy central load of the fully centralized structure as well as the disadvantages of high individual capability requirements, communication complexity, and the command conflicts of the fully distributed structure. The structure conforms to the actual combat scenario as well as the development status of UAV swarm technology and will become more normalized and practical [29; 30; 31; 32]. The algebraic graph theory is used to describe the internal interaction of the UAV swarm system. Assume that the UAV swarm has \(N\) UAVs, and each UAV is regarded as a node, then the communication relationship is seen as an edge. A directed graph \(G=\{V,E,W\}\) which consists of the node set \(V=\{v_{1},v_{2},\ \ldots,\,v_{N}\}\), the edge set \(E\subseteq\{(v_{i},v_{j}):v_{i},v_{j}\in V,i\
eq j\}\) and the adjacency matrix \(\mathbf{W}=[w^{ij}]\in\mathbb{R}^{N\times N}\), with non-negative entries \(w^{ij}\). The entries in \(\mathbf{W}\) are defined with \(w^{ij}=1\) for \((v_{i},v_{j})\in E\) and \(w^{ij}=0\) otherwise. In addition, \(w^{ii}=0\) for all \(i=1,2,\cdots,N\). The neighbor set of node \(v_{i}\) is described as \(N_{i}=\{v_{j}:(v_{i},v_{j})\in E\}\). There is a top leader, \(N_{m}\) group leaders, and \(N_{f}\) followers in UAV swarms with distributed subsets, as shown in Figure 1. Each subset \(i\) has \(N_{h}\) followers. The top leader is the highest leader node and leads the initial task assignment. The group leader, which obtains information of each UAV in the subset, is the leader node of the subset, participates in target allocation on behalf of the subset, and leads subtask assignment; the followers are members of the subset and participate in subtask assignment and execution. Each node is numbered in the UAV swarm, in which the top leader index is 1, the indexes of group leaders are \(i=2,\ \ldots,\ N_{m}+1\), and the indices of followers are \(i=N_{m}+2\), \(\ldots,\ N_{m}+N_{f}+1\). The adjacency matrix \(W\in\mathbb{R}^{N\times N}\) has the following form:  
\[\mathbf{W}=\left[ \begin{array}{ccc}0&\mathbf{W}^{TM}&0\\ \mathbf{W}^{MT}&\mathbf{W}^{M0}&\mathbf{W}^{MF}\\ 0&\mathbf{W}^{FM}&\mathbf{W}^{F0}\end{array} \right]_{N\times N} \tag{1}\]  
where \(\mathbf{W}^{MT}\in\mathbb{R}^{N_{m}\times 1}\) and \(\mathbf{W}^{TM}\in\mathbb{R}^{1\times N_{m}}\) indicate the communication topology among the top leader and group leaders, \(\mathbf{W}^{M0}\in\mathbb{R}^{N_{m}\times N_{m}}\) expresses the communication topology among group leaders, the communication topology among group leaders, and their follow ers are denoted as \(\mathbf{W}^{\text{M0}}\in\mathbb{R}^{N_{m}\times N_{m}}\), while \(\mathbf{W}^{\text{F0}}\in\mathbb{R}^{N_{f}\times N_{f}}\) is denoted as the communication topology among followers. Assuming that no direct communication exists between each follower and the top leader, the followers of each subset only communicate within UAVs in the subset:  
\[ \begin{array}{l}\mathbf{W}^{FM}=diag\big{\{}\mathbf{W}^{FM_{1}},\mathbf{W}^{FM_{2}}, \ldots,\mathbf{W}^{FM_{N_{m}}}\big{\}}\\ \mathbf{W}^{\text{MF}}=diag\big{\{}\mathbf{W}^{\text{MF}_{1}},\mathbf{W}^{\text{MF}_{2}}, \ldots,\mathbf{W}^{\text{MF}_{N_{m}}}\big{\}}\\ \mathbf{W}^{\text{F0}}=diag\big{\{}\mathbf{W}^{\text{F1}},\mathbf{W}^{\text{F2}},\ldots, \mathbf{W}^{\text{F3}_{N_{m}}}\big{\}}\end{array}