Maintaining high system performance is critical for a multi-hop flying mesh network (FlyMesh) to perform missions in different environments. Although the Poisson point process (PPP) has been widely used for the performance analysis of FlyMesh, it still has flaws in describing the spatial distribution of the UAVs since it does not restrict the minimum distance between them. The spatial deployment of FlyMesh varies depending on the environment. Considering the relevance and practicality, we modeled the multi-hop FlyMesh using the $\beta$-Ginibre point process ($\beta$-GPP) and equipped each UAV with a directional antenna. Under the condition of the decode-and-forward protocol, we derived the connection probability and ergodic capacity of a multi-hop FlyMesh utilizing the Laplace transform of interference. Then, we calculated an approximate expression for the interference Laplace transform based on the diagonal approximation and further obtained the coverage probability. Finally, the numerical simulation results verified the correctness of the theoretical derivation, indicating that it is possible to optimize the system’s performance based on the expressions derived in this paper.