Graph Neural Networks Over the Air for Decentralized Tasks in Wireless Networks

Graph neural networks (GNNs) model representations from networked data and allow for decentralized execution through localized communications. Existing GNNs 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 and permits decentralized execution with over-the-air computation. AirGNNs modify graph convolutional operations that shift graph signals over random communication graphs to account for channel fading and noise when aggregating features from neighbors, thus, improving architecture robustness to channel impairments. 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, multi-robot flocking and wireless channel management corroborate theoretical findings and show superior performance of AirGNNs over wireless communication channels.