HeroFilter: Adaptive Spectral Graph Filter for Varying Heterophilic Relations

TL;DR

This work tackles graph heterophily by showing that fixed low-pass or high-pass spectral filters are insufficient across diverse graphs due to a non-monotonic relationship between heterophily and spectral response. It introduces HeroFilter, an adaptive spectral filtering framework composed of a Patcher that selects spectrally relevant neighbors and a Mixer that jointly aggregates patches across spatial and feature dimensions, with a scalable Fast-HeroFilter variant that avoids eigen-decomposition. The authors provide a theoretical bound linking graph heterophily, spectral processing, and generalization, and demonstrate state-of-the-art or competitive performance across 16 datasets, including large-scale graphs, with up to 9.2% accuracy gains. The framework bridges spectral GNN insights and practical scalability, offering an interpretable, architecture-agnostic approach for robust graph representations in both homophilic and heterophilic regimes.

Abstract

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 G, how and to what extent will the varying heterophily degree of 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 [METHOD NAME], a simple yet powerful GNN, which extracts information across the heterophily spectrum and combines salient representations through adaptive mixing. [METHOD NAME]'s superior performance achieves up to 9.2% accuracy improvement over leading baselines across homophilic and heterophilic graphs.

Figures (5)

• Figure 1: 3D visualization showing the underlying relationship among eigenvalues, heterophily degree, and frequency response (in log scale). In particular, we synthesized nine graphs with heterophily degrees ranging from 0.1 to 0.9. Each curve with a unique color corresponds to a well-trained spectral graph filter tang2019chebnet on each graph.
• Figure 2: An interpolated figure demonstrating the relationship between filtered eigenvalues, heterophily degree, and accuracy.
• Figure 3: Overview of the HeroFilter framework, consisting of (1) the HeroFilter Patcher, which adaptively selects spectrally relevant neighbors using learned filters, and (2) the HeroFilter Mixer, which aggregates patch features across nodes and dimensions.
• Figure 4: Parameter sensitivity analysis results for Cora (left) and Texas (right) datasets, respectively.
• Figure :

Theorems & Definitions (11)

• Definition 1: Node Heterophily
• Definition 2: Heterophily Degree Vector in Spectral Domain
• Proposition 1
• Proposition 2
• Theorem 1
• Proposition 2
• proof
• Proposition 2
• proof
• Theorem 1