# Topology Duration Optimization for UAV Swarm Network under the System Performance Constraint
## 3 System Model
### Problem Formulation  
In this paper, the object is to find an optimal \(\mathbf{G}_{d}(t)\in\mathbf{G}(t)\) that maximize the topology duration of the swarm network. Meanwhile, for a UAV swarm system, end-to-end transmission delay, network throughput, and energy consumption have to be comprehensively considered. Therefore, the problem of network construction for a UAV swarm can be preliminarily formulated as  
\[\arg \max\mathrm{T}(\mathbf{G}_{d}(t))\] (4) s.t. \[\mathbf{G}_{d}(t)\in\mathbf{G}(t) \tag{5}\] \[\sigma_{ij}\leq\sigma_{th},i\
eq j\text{ and }i,j\in[1,N]\] (6) \[M\geq M_{th}\] (7) \[E_{i}^{c}\leq E_{th}^{c},i\in[1,N] \tag{8}\]  
In the object function of Equation (4), \(\mathrm{T}(\cdot)\) denotes the topology duration operator. Equation (5) indicates that the optimal \(\mathbf{G}_{d}(t)\) is a subset of \(\mathbf{G}(t)\). Equations (6)-(8), respectively, ensure that the performance of end-to-end transmission delay, network throughput, and energy consumption must meet the requirements of the system. \(\sigma_{ij}\) is the end-to-end delay between node \(i\) and node \(j\), \(M\) is network overall throughput, and \(E_{i}^{c}\) is the energy consumption for data transmission of node \(i\), while \(\sigma_{th}\), \(M_{th}\) and \(E_{th}^{c}\) are corresponding performance thresholds that should be satisfied when obtaining the optimal network topology. In practice, the values of \(\sigma_{th}\), \(M_{th}\) and \(E_{th}^{c}\) must determined according to specific scenario and system requirements. For this model, it is not an easy task to directly find the optimal solution, especially with the increase of nodes number \(N\). Thus, we need to reformulate it according to the characteristics of UAV swarm systems, which are analyzed as follows. Firstly, lower end-to-end delay leads to an enhancement of network interaction efficiency, collaborative tasks performance and the swarm operation security [28; 29; 30]. In general, the end-to-end delay for a UAV swarm network is mainly consisted of wireless transmission delay, multi-hop forwarding delay and signal processing delay. Among them, the multi-hop forwarding delay is the critical factor that severely affects the overall end-to-end delay, and it is closely related to the design of network topology construction. Therefore, the end-to-end delay can be constrained by the number of forwarding hops for data transmission between any two nodes in the UAV swarm network, which is represented as  
\[hop_{ij}\leq hop_{\max} \tag{9}\]  
Figure 2: A schematic diagram for the universal set \(\mathbf{G}(t)\) and typical topology \(\mathbf{G}_{d}(t)\). (**a**) The universal set of network topology \(\mathbf{G}(t)\); (**b**) a typical \(\mathbf{G}_{d}(t)\) based on topology design. where \(hop_{ij}\) is the number of hops between node \(i\) and \(j\), while \(hop_{\max}\) is maximum number of hops allowed on the swarm network. Secondly, under a particular physical-layer design, the throughput performance of swarm network is mainly depends on the relative distance between nodes and the number of forwarding hops in the link. Relative distance affects the adaptive modulation and coding strategy between communication nodes [31; 32; 33], while forwarding hops has an important influence on the bit transmission rate of links [34; 35]. Since the number of forwarding hops have been constrained in Equation (9), here we focus on the condition of relative distance. Thus, the condition of Equation (7) can be converted to average distance constraint between nodes, which is presented as  
\[\overline{d_{ij}}\leq d_{th} \tag{10}\]  
where \(\overline{d_{ij}}\) is the average distance between nodes in the constructed network topology and \(d_{th}\) is the threshold. Thirdly, the node energy consumption is one of the key factors in the design, which has an important influence on the performance of swarm system.