Graph neural networks (GNNs) model representations from networked data and allow for decentralized inference through localized communications. Existing GNN architectures often assume ideal communications and ignore potential channel effects, such as fading and noise, leading to performance degradation in real-world implementation. Considering a GNN implemented over nodes connected through wireless links, this paper conducts a stability analysis to study the impact of channel impairments on the performance of GNNs, and proposes graph neural networks over the air (AirGNNs), a novel GNN architecture that incorporates the communication model. AirGNNs modify graph convolutional operations that shift graph signals over random communication graphs to take into account channel fading and noise when aggregating features from neighbors, thus, improving architecture robustness to channel impairments during testing. We develop a channel-inversion signal transmission strategy for AirGNNs when channel state information (CSI) is available, and propose a stochastic gradient descent based method to train AirGNNs when CSI is unknown. The convergence analysis shows that the training procedure approaches a stationary solution of an associated stochastic optimization problem and the variance analysis characterizes the statistical behavior of the trained model. Experiments on decentralized source localization and multi-robot flocking corroborate theoretical findings and show superior performance of AirGNNs over wireless communication channels.

翻译：图神经网络（GNN）从网络化数据中建模表征，并可通过局部通信实现分散推理。现有GNN架构通常假设理想通信，忽略信道效应（如衰落和噪声），导致在实际部署中性能下降。针对通过无线链路连接的节点实现的GNN，本文通过稳定性分析研究信道损伤对GNN性能的影响，并提出空口图神经网络（AirGNNs）——一种融合通信模型的新型GNN架构。AirGNNs改进了图卷积操作（该操作在随机通信图上平移图信号），在聚合邻居特征时考虑信道衰落与噪声，从而提升架构对测试阶段信道损伤的鲁棒性。我们为AirGNNs设计了信道反转信号传输策略（当信道状态信息CSI可用时），并提出了基于随机梯度下降的方法以在CSI未知时训练AirGNNs。收敛性分析表明训练过程趋近于相关随机优化问题的平稳解，方差分析则刻画了训练模型的统计特性。在分散源定位与多机器人集群实验中的结果验证了理论分析，并展现了AirGNNs在无线通信信道上的优越性能。