Rethinking Graph Neural Networks From A Geometric Perspective Of Node Features

Many works on graph neural networks (GNNs) focus on graph topologies and analyze graph-related operations to enhance performance on tasks such as node classification. In this paper, we propose to understand GNNs based on a feature-centric approach. Our main idea is to treat the features of nodes from each label class as a whole, from which we can identify the centroid. The convex hull of these centroids forms a simplex called the feature centroid simplex, where a simplex is a high-dimensional generalization of a triangle. We borrow ideas from coarse geometry to analyze the geometric properties of the feature centroid simplex by comparing them with basic geometric models, such as regular simplexes and degenerate simplexes. Such a simplex provides a simple platform to understand graph-based feature aggregation, including phenomena such as heterophily, oversmoothing, and feature re-shuffling. Based on the theory, we also identify simple and useful tricks for the node classification task.