A Neural Collapse Perspective on Feature Evolution in Graph Neural Networks

TL;DR

This paper investigates how features evolve in graph neural networks during training, using the Neural Collapse phenomenon, and provides empirical and theoretical insights into the conditions for feature collapse and their implications for GNN generalization.

Contribution

It introduces a neural collapse perspective to analyze feature evolution in GNNs, combining empirical observations with theoretical models to explain partial collapse and graph structural conditions.

Findings

Within-class variability decreases in node classification, but less than in image classification.

Strict graph structural conditions are needed for exact feature collapse in models.

Partial collapse can be explained by gradient dynamics and spectral analysis.

Abstract

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.