Graph Neural Networks (GNNs) have achieved great success but are often considered to be challenged by varying levels of homophily in graphs. Recent empirical studies have surprisingly shown that homophilic GNNs can perform well across datasets of different homophily levels with proper hyperparameter tuning, but the underlying theory and effective architectures remain unclear. To advance GNN universality across varying homophily, we theoretically revisit GNN message passing and uncover a novel smoothness-generalization dilemma, where increasing hops inevitably enhances smoothness at the cost of generalization. This dilemma hinders learning in high-order homophilic neighborhoods and all heterophilic ones, where generalization is critical due to complex neighborhood class distributions that are sensitive to shifts induced by noise or sparsity. To address this, we introduce the Inceptive Graph Neural Network (IGNN) built on three simple yet effective design principles, which alleviate the dilemma by enabling distinct hop-wise generalization alongside improved overall generalization with adaptive smoothness. Benchmarking against 30 baselines demonstrates IGNN's superiority and reveals notable universality in certain homophilic GNN variants. Our code and datasets are available at https://github.com/galogm/IGNN.

Heterophilous graphs, where dissimilar nodes tend to connect, pose a challenge for graph neural networks (GNNs). Increasing the GNN depth can expand the scope (i.e., receptive field), potentially finding homophily from the higher-order neighborhoods. However, GNNs suffer from performance degradation as depth increases. Despite having better expressivity, state-of-the-art deeper GNNs achieve only marginal improvements compared to their shallow variants. Through theoretical and empirical analysis, we systematically demonstrate a shift in GNN generalization preferences across nodes with different homophily levels as depth increases. This creates a disparity in generalization patterns between GNN models with varying depth. Based on these findings, we propose to improve deeper GNN generalization while maintaining high expressivity by Mixture of scope experts at test (Moscat). Experimental results show that Moscat works flexibly with various GNNs across a wide range of datasets while significantly improving accuracy. Our code is available at https://github.com/Hydrapse/moscat.

Deep neural networks (DNNs) have been shown to memorize their training data, but similar analyses for graph neural networks (GNNs) remain under-explored. We introduce NCMemo(Node Classification Memorization), the first framework to quantify label memorization in semi-supervised node classification. We establish an inverse relationship between memorization and graph homophily, i.e., the tendency of connected nodes to share labels or features. Lower homophily significantly increases memorization, indicating that GNNs rely on label memorization when learning less homophilic graphs. We then analyze GNN training dynamics and find that increased memorization in low-homophily graphs is tightly coupled to GNNs' implicit bias toward using graph structure.

A.1 Proof of Theorem 1 In this proof, we adopt a simplified version of our message-passing function that ignores the skipconnection: The HGNN trained in the experimental results shown in Figure 2 also does not use skip-connections and hence represents a theoretically-exact KTN component. In the real experiments, we use (1) skip-connections, exploiting their usual benefits (12), and (2) the trainable version of KTN. Without loss of generality, we prove the result for the case where R = {(s,t): s,t T }, meaning the type of an edge is identified with the (ordered) types of the neighbor nodes. In other words, there is only one edge modality possible, such as a social networks with multiple node types (e.g. "friendship" and "message"), the result is extended trivially (through with more algebraically-dense forms of ats and qts). The output of Aggregate is a concatenation of edge-type-specific aggregations (see Equation 3).

Graph Neural Networks (GNNs) have received extensive research attention for their promising performance in graph machine learning. Despite their extraordinary predictive accuracy, existing approaches, such as GCN and GPRGNN, are not robust in the face of homophily changes on test graphs, rendering these models vulnerable to graph structural attacks and with limited capacity in generalizing to graphs of varied homophily levels. Although many methods have been proposed to improve the robustness of GNN models, the majority of these techniques are restricted to the spatial domain and employ complicated defense mechanisms, such as learning new graph structures or calculating edge attention. In this paper, we study the problem of designing simple and robust GNN models in the spectral domain. We propose EvenNet, a spectral GNN corresponding to an even-polynomial graph filter. Based on our theoretical analysis in both spatial and spectral domains, we demonstrate that EvenNet outperforms full-order models in generalizing across homophilic and heterophilic graphs, implying that ignoring odd-hop neighbors improves the robustness of GNNs. We conduct experiments on both synthetic and real-world datasets to demonstrate the effectiveness of EvenNet. Notably, EvenNet outperforms existing defense models against structural attacks without introducing additional computational costs and maintains competitiveness in traditional node classification tasks on homophilic and heterophilic graphs.

Homophily is a graph property describing the tendency of edges to connect similar nodes; the opposite is called heterophily. It is often believed that heterophilous graphs are challenging for standard message-passing graph neural networks (GNNs), and much effort has been put into developing efficient methods for this setting. However, there is no universally agreed-upon measure of homophily in the literature. In this work, we show that commonly used homophily measures have critical drawbacks preventing the comparison of homophily levels across different datasets. For this, we formalize desirable properties for a proper homophily measure and verify which measures satisfy which properties.

Graph homophily refers to the phenomenon that connected nodes tend to share similar characteristics. Understanding this concept and its related metrics is crucial for designing effective Graph Neural Networks (GNNs). The most widely used homophily metrics, such as edge or node homophily, quantify such similarity as label consistency across the graph topology. These metrics are believed to be able to reflect the performance of GNNs, especially on node-level tasks. However, many recent studies have empirically demonstrated that the performance of GNNs does not always align with homophily metrics, and how homophily influences GNNs still remains unclear and controversial.