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

TL;DR

A stability analysis is conducted to study the impact of channel impairments on the performance of GNNs, and a novel GNN architecture that incorporates the communication model and permits decentralized execution with over-the-air computation is proposed.

Abstract

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.

Figures (5)

• Figure 1: The filter frequency response over the air with $K=2$. The function $f(\boldsymbol{\lambda})$ is determined by the filter parameters $\{ \alpha_k \}_{k=0}^K$ and independent of the underlying graph. For two specific sequences of AirGSOs, $h(\boldsymbol{\lambda})$ is instantiated on two specific sets of eigenvalues in blue and green.
• Figure 2: (a) Training procedures of the AirGNNs with and without CSI for $\beta=1$. (b) Performance of the AirGNN and baselines in communication scenarios with different scale parameters $\beta$ for source localization. (c) Performance of the AirGNN and baselines in communication scenarios with different SNRs for source localization.
• Figure 3: (a) Performance of the AirGNNs with different power cut-off thresholds $\gamma$ in different channel conditions. (b) Performance of the AirGNNs with different numbers of features $F$. (c) Performance of the AirGNNs with different graph sizes, i.e., numbers of nodes $n$.
• Figure 4: (a) Performance of the AirGNNs and baselines in communication scenarios with different scale parameters $\beta$ for multi-robot flocking. (b) Performance of the AirGNNs and baselines in communication scenarios with different SNRs for multi-robot flocking. (c) Performance of the AirGNNs with different power cut-off thresholds $\gamma$. (d) Zoomed version of Fig. \ref{['subfig:flockingGamma']}.
• Figure 5: Performance of the GNN under channel impairments in different communication scenarios. (a) Different means $\mu$. (b) Different standard deviations $\delta$. (c) Different SNRs.

Theorems & Definitions (7)

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