  Graph neural networks (GNNs) have become increasingly popular for
classification tasks on graph-structured data. Yet, the interplay between graph
topology and feature evolution in GNNs is not well understood. In this paper,
we focus on node-wise classification, illustrated with community detection on
stochastic block model graphs, and explore the feature evolution through the
lens of the "Neural Collapse" (NC) phenomenon. When training instance-wise deep
classifiers (e.g. for image classification) beyond the zero training error
point, NC demonstrates a reduction in the deepest features' within-class
variability and an increased alignment of their class means to certain
symmetric structures. We start with an empirical study that shows that a
decrease in within-class variability is also prevalent in the node-wise
classification setting, however, not to the extent observed in the
instance-wise case. Then, we theoretically study this distinction.
Specifically, we show that even an "optimistic" mathematical model requires
that the graphs obey a strict structural condition in order to possess a
minimizer with exact collapse. Interestingly, this condition is viable also for
heterophilic graphs and relates to recent empirical studies on settings with
improved GNNs' generalization. Furthermore, by studying the gradient dynamics
of the theoretical model, we provide reasoning for the partial collapse
observed empirically. Finally, we present a study on the evolution of within-
and between-class feature variability across layers of a well-trained GNN and
contrast the behavior with spectral methods.
