A Spectral Analysis of Graph Neural Networks on Dense and Sparse Graphs

In this work we propose a random graph model that can produce graphs at different levels of sparsity. We analyze how sparsity affects the graph spectra, and thus the performance of graph neural networks (GNNs) in node classification on dense and sparse graphs. We compare GNNs with spectral methods known to provide consistent estimators for community detection on dense graphs, a closely related task. We show that GNNs can outperform spectral methods on sparse graphs, and illustrate these results with numerical examples on both synthetic and real graphs.