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图神经网络（GNN）在处理深层架构时，长期面临两大根本性挑战：一是过平滑（over-smoothing），即随着网络层数加深，不同节点的表征趋于同质化、难以区分；二是过挤压（over-squashing），即远距离信息在有限的信息传递通道中遭到严重压缩。尽管诸如狄利克雷能量（Dirichlet energy）等现有的全局指标有助于量化过平滑现象，但它们缺乏节点级别的分辨率，无法有效指导针对性的架构优化。

为了填补这一空白，本文引入了 LEED（Local Embedding Evolution Distance，局部嵌入演化距离）——一种新颖的局部指标，能够追踪各个节点在不同层间嵌入轨迹的演化。LEED 实现了对异质过平滑模式的细粒度分析，并推导出了强健的、由嵌入驱动的节点中心性得分。通过将 LEED 作为构建局部虚拟节点（Local Virtual Nodes）的单一评判标准，作者提出了一种更高效、更具针对性的策略，能够有效缓解过挤压并提升 GNN 在各类数据集上的整体性能。

LEED: Local Embedding Evolution Distance for Over-smoothing Estimation and Virtual Node Selection in GNN

LEED: Local Embedding Evolution Distance for Over-smoothing Estimation and Virtual Node Selection in GNN

Summary

Summary

Graph Neural Networks (GNNs) traditionally struggle with over-smoothing (where node representations become indistinguishable across deep layers) and over-squashing (where long-range information is heavily compressed). While existing global metrics like Dirichlet energy help quantify over-smoothing, they lack the node-level resolution needed to guide targeted architectural enhancements.

Graph Neural Networks (GNNs) traditionally struggle with over-smoothing (where node representations become indistinguishable across deep layers) and over-squashing (where long-range information is heavily compressed). While existing global metrics like Dirichlet energy help quantify over-smoothing, they lack the node-level resolution needed to guide targeted architectural enhancements.

To bridge this gap, this paper introduces LEED (Local Embedding Evolution Distance)—a novel local metric that tracks individual node embedding trajectories across layers. LEED enables fine-grained analysis of heterogeneous over-smoothing patterns and derives robust, embedding-driven node centrality scores. By utilizing LEED as a singular criterion for constructing Local Virtual Nodes, the authors present a more efficient and effective strategy to mitigate over-squashing and boost GNN performance across datasets.

To bridge this gap, this paper introduces LEED (Local Embedding Evolution Distance)—a novel local metric that tracks individual node embedding trajectories across layers. LEED enables fine-grained analysis of heterogeneous over-smoothing patterns and derives robust, embedding-driven node centrality scores. By utilizing LEED as a singular criterion for constructing Local Virtual Nodes, the authors present a more efficient and effective strategy to mitigate over-squashing and boost GNN performance across datasets.

Document Metadata

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Field Details
Title LEED: Local Embedding Evolution Distance for over-smoothing estimation and virtual node selection in GNN
Authors Killian Cressant, Pedro B. Velloso
Subjects Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv ID arXiv:2608.09596
DOI 10.48550/arXiv.2608.09596
Submission History • v1: August 10, 2026
• v2: September 4, 2026 (This version)
Field Details
Title LEED: Local Embedding Evolution Distance for over-smoothing estimation and virtual node selection in GNN
Authors Killian Cressant, Pedro B. Velloso
Subjects Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv ID arXiv:2608.09596
DOI 10.48550/arXiv.2608.09596
Submission History • v1: August 10, 2026
• v2: September 4, 2026 (This version)

Abstract

Abstract

图神经网络（GNN）面临着两个根本性的局限性：一是过平滑，即节点表征随着网络深度的增加而变得无法区分；二是过挤压，即远距离信息通过有限的信息传递通道被压缩。现有的诸如狄利克雷能量（Dirichlet energy）等指标提供了对过平滑的全局表征，但缺乏分析节点级行为和指导架构改进所需的空间分辨率。

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.

在本文中，我们提出了 LEED（局部嵌入演化距离，Local Embedding Evolution Distance），这是一种新颖的局部指标，通过追踪各层中单个节点嵌入的演化来量化过平滑。通过在节点级别上运行，LEED 能够在训练过程中对表征动态进行细粒度分析，揭示出全局基于能量的度量所无法察觉的异质过平滑模式。这种局部性衍生出了具有信息量的节点重要性得分，可被解释为嵌入驱动的中心性度量。我们利用 LEED 设计了一种更高效的虚拟节点选择策略。与依赖多种启发式中心性度量的现有方法不同，我们的方法使用 LEED 作为唯一准则来指导局部虚拟节点（Local Virtual Nodes）的构建，从而缓解过挤压问题。实验表明，在保持全局评估能力的同时，LEED 提供了比狄利克雷能量更具信息量的诊断结果，并实现了更有效的虚拟节点集成，从而提升了 GNN 在各个数据集上的性能。

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.

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