  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.
