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Unmanned aerial vehicles (UAVs) can be deployed as base stations (BSs) for emergency communications of user equipments (UEs) in 5G/6G networks. In multi-UAV communication networks, UAVs' load balancing and UEs' data rate fairness are two challenging problems and can be optimized by UAV deployment strategies. In this work, we found that these two problems are related by the same performance metric, which makes it possible to optimize the two problems simultaneously. To solve this joint optimization problem, we propose a UAV diffusion deployment algorithm based on the virtual force field method. Firstly, according to the unique performance metric, we define two new virtual forces, which are the UAV-UAV force and UE-UAV force defined by FU and FV, respectively. FV is the main contributor to load balancing and UEs' data rate fairness, and FU contributes to fine tuning the UEs' data rate fairness performance. Secondly, we propose a diffusion control stratedy to the update UAV-UAV force, which optimizes FV in a distributed manner. In this diffusion strategy, each UAV optimizes the local parameter by exchanging information with neighbor UAVs, which achieve global load balancing in a distributed manner. Thirdly, we adopt the successive convex optimization method to update FU, which is a non-convex problem. The resultant force of FV and FU is used to control the UAVs' motion. Simulation results show that the proposed algorithm outperforms the baseline algorithm on UAVs' load balancing and UEs' data rate fairness. [ABSTRACT FROM AUTHOR]