## 1 Introduction  
Recently, unmanned aerial vehicles (UAVs) have received much attention and research due to their controlled mobility and on-demand deployment. For example, UAVs can be used as airborne wireless communication platforms, such as mobile base stations, to quickly restore communication services and ensure communication quality [1]. UAVs can also act as mobile relays to connect two or more distant users for communication [2]. As an aerial base station or mobile relay, they greatly improve the service quality of the network [3]. In addition, UAVs can be applied to mobile data collection and information dissemination to facilitate various Internet of Things (IoT) applications. Multiple UAVs are organized into a group and deployed together to achieve the coverage of an area, in which UAVs communicate with each other in a decentralized and ad hoc manner; this type of network is called a flying mesh network (FlyMesh). Directional antennas are used to improve the efficient coverage of UAVs. At the same time, they can increase the network capacity and increase the robustness of the network [4]. It is known that FlyMesh will experience frequent topological changes due to the rapid movement of UAV nodes [5]. To ensure the continuous service of FlyMesh, maintenance personnel must maintain good connections between drones. Therefore, establishing a suitable mathematical model to analyze the performance of FlyMesh is crucial. Stochastic geometry is commonly used as a tool for modeling and analyzing the performance of FlyMesh [6]. For ease of handling, UAV locations are usually modeled as a Poisson point process (PPP) [7]. PPP only applies to networks where transmitters(jammers) are deployed in a completely unplanned manner, where each UAV is considered an independent point in space and there is no relationship between UAV locations. However, existing models are difficult to reconcile because UAVs usually keep a distance from each other to improve coverage and prevent collisions. In addition, UAVs commonly use directional antennas to reduce self-interference of signals. However, in real-world environments, the location of UAVs is carefully planned to avoid interference or to extend the coverage area. That is to say, there is a repulsion (or attraction) between UAVs, which means that the actual deployed UAV locations usually form a more regular pattern of points than PPP. The \(\beta\)-Ginibre point process (\(\beta\)-GPP) has received wide attention as point processes reflect repulsive forces [8]. \(\beta\)-GPP is repulsive and treats PPP as a particular case. In this context, \(\beta\)-GPP was utilized to explain the spatial regularity among UAVs in this paper. In contrast to PPP, \(\beta\)-GPP employs the parameter \(\beta\) to restrict the minimum distance between UAVs. Different deployment environments are reflected by the repulsive parameter \(\beta\). GPP is similar to PPP when \(\beta\to 0\), and the UAVs are in a uniform distribution while \(\beta\to\)1. Therefore, the distribution of UAVs in a given environment can be more accurately described by the repulsive parameter \(\beta\). We derived the connection probability of a multi-hop relay link by utilizing the Laplace function of the \(\beta\)-GPP distribution, based on the connection probability to derive the ergodic capacity of the multi-hop relay. To better describe the ability of FlyMesh to provide services, we used the approximate expression of the diagonal kernel matrix to calculate the coverage probability. Finally, numerical simulations verified the accuracy of the derived theoretical expressions. The main objective of this paper was to introduce and promote \(\beta\)-GPP as a UAV network model in which nodes exhibit repulsive forces. Compared with the PPP model, \(\beta\)-GPP not only captures the application scenarios of real networks but is also analytically easy to handle.