  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.
