Hierarchical Multi-Scale Graph Neural Networks: Scalable Heterophilous Learning with Oversmoothing and Oversquashing Mitigation

2026-05-08

AI Summary

This work addresses key limitations of existing spectral graph neural networks for heterophilous graphs, which suffer from dominant hub nodes, oversmoothing, over-squashing, and approximation errors inherent in polynomial filters. The authors propose HMH, a novel framework that uniquely integrates hierarchical sparse orthogonal Haar bases with heterophily-aware embeddings. Specifically, a heterophily-aware encoder learns signed affinities to construct a soft hierarchical graph structure, wherein learnable spectral filters are applied at each level. Multi-scale information is effectively fused through skip-connection-based unpooling. This approach substantially mitigates the aforementioned issues, achieving up to a 3% absolute improvement in node classification and a 7-percentage-point gain in graph classification over state-of-the-art spectral methods, while maintaining near-linear scalability.

Abstract

Graphs with heterophily, where adjacent nodes carry different labels, are prevalent in real-world applications, from social networks to molecular interactions. However, existing spectral Graph Neural Network (GNN) approaches tailored for heterophilous graph classification suffer from hub-dominated (node with large degree) aggregation and oversmoothing, as their suboptimal polynomial filters introduce approximation errors and blend distant signals. To address the degree-biased aggregation and suboptimal polynomial filtering, we introduce a Hierarchical Multi-view HAAR (HMH), a novel spectral graph-learning framework that scales in near-linear time . HMH first learns feature- and structure-aware signed affinities via a heterophily-aware encoder, then constructs a soft graph hierarchy guided by these embeddings. At each hierarchical level, HMH constructs a sparse, orthonormal, and locality-aware Haar basis to apply learnable spectral filters in the frequency domain. Finally, skip-connection unpooling layers combine outputs from all hierarchical levels back into the original graph, effectively preventing hub domination and long-range signal bottleneck (over-squashing). Experimentation shows that HMH outperforms state-of-the-art spectral baselines, achieving up to a 3% improvement on node classification and 7% points on graph classification datasets, all while maintaining linear scalability.

Problem

• heterophily
• oversmoothing
• oversquashing
• graph neural networks
• spectral filtering

Innovation

• heterophilous graphs
• spectral graph neural networks
• Haar basis
• oversmoothing mitigation
• oversquashing mitigation