HeroFilter: Adaptive Spectral Graph Filter for Varying Heterophilic Relations

Graph heterophily, where connected nodes have different labels, has attracted significant interest recently. Most existing works adopt a simplified approach - using low-pass filters for homophilic graphs and high-pass filters for heterophilic graphs. However, we discover that the relationship between graph heterophily and spectral filters is more complex - the optimal filter response varies across frequency components and does not follow a strict monotonic correlation with heterophily degree. This finding challenges conventional fixed filter designs and suggests the need for adaptive filtering to preserve expressiveness in graph embeddings. Formally, natural questions arise: Given a heterophilic graph $\mathcal{G}$ , how and to what extent will the varying heterophily degree of $\mathcal{G}$ affect the performance of GNNs? How can we design adaptive filters to fit those varying heterophilic connections? Our theoretical analysis reveals that the average frequency response of GNNs and graph heterophily degree do not follow a strict monotonic correlation, necessitating adaptive graph filters to guarantee good generalization performance. Hence, we propose HeroFilter, a simple yet powerful GNN, which extracts information across the heterophily spectrum and combines salient representations through adaptive mixing. HeroFilter's superior performance achieves up to 9.2% accuracy improvement over leading baselines across homophilic and heterophilic graphs.