LEED: Local Embedding Evolution Distance for over-smoothing estimation and virtual node selection in GNN

2026-08-10

AI Summary

Graph neural networks often suffer from oversmoothing and overcompression, yet existing global metrics struggle to characterize representation degradation at the node level. To address this, this work proposes LEED—a node-level local metric that fine-grainedly quantifies oversmoothing by tracking the embedding evolution distance of each node across layers. This metric further serves as a unified guide for constructing local virtual nodes to mitigate overcompression. Departing from multiple heuristic centrality strategies, LEED uniquely adopts embedding evolution distance as a unified criterion for both diagnosis and optimization. It preserves global assessment capability while enabling precise, node-wise diagnostics, leading to significant performance improvements across multiple benchmark datasets.

Abstract

Graph Neural Networks (GNNs) suffer from two fundamental limitations: over-smoothing, where node representations become indistinguishable with depth, and over-squashing, where long-range information is compressed through limited message-passing channels. Existing metrics such as Dirichlet energy provide global characterizations of over-smoothing but lack the resolution to analyze node-level behavior and guide architectural improvements. In this paper, we propose LEED (Local Embedding Evolution Distance), a novel local metric that quantifies over-smoothing by tracking the evolution of individual node embeddings across layers. By operating at the node level, LEED enables fine-grained analysis of representation dynamics during training, revealing heterogeneous over-smoothing patterns that are invisible to global energy-based measures. This locality induces informative node importance scores, interpreted as embedding-driven centrality measures. We leverage LEED to design a more efficient strategy for virtual node selection. Unlike existing approaches that depend on multiple heuristic centrality measures, our method uses LEED as a unique criterion to guide the construction of Local Virtual Nodes to mitigate over-squashing. Experiments show that LEED provides more informative diagnostics than Dirichlet energy while preserving global evaluation, and enables more effective virtual node integration, improving GNN performance across datasets.

Problem

• over-smoothing
• over-squashing
• Graph Neural Networks
• virtual node selection
• node-level analysis

Innovation

• LEED
• over-smoothing
• virtual node selection
• local metric
• graph neural networks