## 2 Theoretical Basis of Time-Varying Target Formation
### Construction of Time-Varying Topology Formation Graphics Library  
In the face of communication loss and other situations in complex environments, if the corresponding graphics are not generated in advance, it will take a lot of precious time to construct and reconstruct the formation, thus will miss the best opportunity. Therefore, if different numbers of target formations that may appear in advance are established, it will save time to call the graphics library when failure occurs at the first time, and a new reconstructed target topology can be efficiently and quickly established. In this section, we first introduce the concept of global rigidity and stress matrix based on graph theory to study the characteristics of formation configuration and describe these characteristics. The corresponding leaders for formation and reconstruction are selected, and the conditions required to construct a time-varying topological formation that meets the requirements are given. Graph theory is a great mathematical tool for simplifying the characterization of multi-agent systems. Moreover, the rigidity [20] will reveal the uniqueness of the formation structure up to congruence. The generic configuration can be expressed by \(q=[q_{1}^{T},\cdots,q_{n}^{T}]^{T}\in R^{nd}\). A structure \((G,q)\) means a corresponding graph \(G\) with its configuration \(q\). A framework \((G,q)\) is rigid in \(R^{d}\), if all the frameworks \((G,p)\) equivalent to \((G,q)\) and sufficiently close to \((G,q)\) are congruent to \((G,q)\) in \(R^{d}\). If all the structures \((G,p)\) equivalent to \((G,q)\) are congruent to \((G,q)\), it is known as globally rigid in \(R^{d}\). Moreover, if this congruent relationship holds in any higher-dimensional space \(R^{\mathbb{D}}\supset R^{d}\), it is universally rigid [21]. In addition, stress is also adopted to describe the robustness and stability of a framework. Essentially, stress means the direction of force with symbols and describes the force per unit length. \(w\) represent the stress of \((G,q)\). It is common to assign a scalar \(w_{ij}=w_{ji}\) for each edge \((i,j)\) of a structure \((G,q)\). Next, \(w\in R^{m}\) refers to the concatenated vector \(w=(\cdots,w_{ij},\cdots)^{T}\). Furthermore, if satisfies  
\[\sum_{j\in N_{i}}w_{ij}(q_{j}-q_{i})=0,\forall i\in\{1,\cdots,n\} \tag{5}\]  
Then, it is defined that \(w\) is equilibrium stress of \((G,q)\). Given \(w\), there exists a stress matrix \(\Omega\in R^{n\times n}\), for which \(\Omega_{ij}=-w_{ij}\) for \(i\
eq j\) and \(\Omega_{ii}=\sum\limits_{i\
eq j}w_{ij}\) for \(i=1,\cdots,n\). In the sequence, Lemmas 1-4 are given. These lemmas lead to conditions on the rank of the matrices that need to be satisfied for affine span as well as generic universal rigidity. In addition, the selection of the leader in the formation and the maximum upper bound on the number of edges are presented. These lemmas give constraints on the design of formation configurations and the establishment of graph libraries. We will introduce Lemma 1 to define the rank condition for affine span. Lemma 1 (Rank condition for affine span): _The set of points \(\{p_{i}\}_{i=1}^{n}\) affinely span\(R^{d}\) if and only if \(n\geq d+1\) andrank\((\overline{P}(p))=d+1\)._  
Lemma 2 [22] illustrates the definition of universal rigidity for a class of frameworks with generic configurations. Lemma 2 (Generic universal rigidity): _Let \((G,q)\) be a generic framework on \(n\) vertices in \(R^{d},d\leq n-2\). Then \((G,q)\) is universally rigid if and only if there exists a positive semi-definite stress matrix \(\Omega\) such that its rank is \(n-d-1\)._  
Lemma 3 (Leader selection for affine localizability): _The nominal formation \((G,r)\) is affinely localizable if and only if \(\{r_{i}\}_{i\in\mathbb{N}_{i}}\) affinely span \(R^{d}\)._  
When there are exactly \(d+1\) leaders, given any leader position \(p_{l}\), there is always a solution \((A,b)\) to Equation (4). When there are more than \(d+1\) leaders, the positions of leaders are interdependent.