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.

As the number of layers in Graph Neural Networks (GNNs) increases, over-smoothing becomes more severe, causing intra-class feature distances to shrink, while heterogeneous representations tend to converge. Most existing methods attempt to address this issue by employing heuristic shortcut mechanisms or optimizing objectives to constrain inter-class feature differences. However, these approaches fail to establish a theoretical connection between message passing and the variation in inter-class feature differences, making it challenging to design methods that target the key influencing factors. To address this gap, this paper first introduces the concept of the separation coefficient, which quantifies the contraction of feature distances between classes during multi-layer message passing. Based on this theory, we propose a low-complexity, pluggable, pseudo-label-based adaptive graph structure adjustment method. This approach effectively enhances the separation coefficient of inter-class features while maintaining intra-class compactness, thereby alleviating the convergence of heterogeneous representations caused by multi-layer aggregation. Experimental results demonstrate that the proposed method significantly improves the discriminability of node representations and enhances node classification performance across various datasets and foundational models.

In node classification using graph neural networks (GNNs), a typical model generates logits for different class labels at each node. A softmax layer often outputs a label prediction based on the largest logit. We demonstrate that it is possible to infer hidden graph structural information from the dataset using these logits. We introduce the key notion of label non-uniformity, which is derived from the Wasserstein distance between the softmax distribution of the logits and the uniform distribution. We demonstrate that nodes with small label non-uniformity are harder to classify correctly. We theoretically analyze how the label non-uniformity varies across the graph, which provides insights into boosting the model performance: increasing training samples with high non-uniformity or dropping edges to reduce the maximal cut size of the node set of small non-uniformity. These mechanisms can be easily added to a base GNN model. Experimental results demonstrate that our approach improves the performance of many benchmark base models.