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Measuring and Relieving the Over-smoothing Problem for Graph Neural Networks from the Topological View
Deli Chen, Yankai Lin, Wei Li, Peng Li, Jie Zhou, Xu Sun
TL;DR
GNN over-smoothing makes nodes from different classes indistinguishable, but the reasons for this problem have been insufficiently explained. This paper quantifies smoothness and over-smoothness, links the issue to information-to-noise ratios shaped by graph topology, and introduces two mitigation methods. Experiments across seven datasets and ten GNN models show that the methods relieve over-smoothing and improve performance, although AdaEdge can make incorrect graph adjustments.
Problem
GNN over-smoothing makes representations of nodes from different classes indistinguishable, while limited study has explained why and how it occurs.
Method
The paper introduces MAD and MADGap to measure smoothness and over-smoothness, then proposes MADReg and AdaEdge to address topology-related information-to-noise imbalance.
Results
Experiments on seven graph datasets and ten typical GNN models show that the proposed methods significantly relieve over-smoothing and improve performance in general cases.
Takeaways & Limitations
Graph topology is a useful target for understanding and alleviating GNN over-smoothing alongside model architecture.
Takeaways & Limitations
AdaEdge still suffers from incorrect graph-adjustment operations, motivating work to reduce them.
Abstract
from arXiv · showhide
Graph Neural Networks (GNNs) have achieved promising performance on a wide range of graph-based tasks. Despite their success, one severe limitation of GNNs is the over-smoothing issue (indistinguishable representations of nodes in different classes). In this work, we present a systematic and quantitative study on the over-smoothing issue of GNNs. First, we introduce two quantitative metrics, MAD and MADGap, to measure the smoothness and over-smoothness of the graph nodes representations, respectively. Then, we verify that smoothing is the nature of GNNs and the critical factor leading to over-smoothness is the low information-to-noise ratio of the message received by the nodes, which is partially determined by the graph topology. Finally, we propose two methods to alleviate the over-smoothing issue from the topological view: (1) MADReg which adds a MADGap-based regularizer to the training objective;(2) AdaGraph which optimizes the graph topology based on the model predictions. Extensive experiments on 7 widely-used graph datasets with 10 typical GNN models show that the two proposed methods are effective for relieving the over-smoothing issue, thus improving the performance of various GNN models.
Introduction
This work quantitatively studies why GNN node representations become over-smoothed and proposes metrics and topology-oriented methods to alleviate the problem. It links over-smoothing to low information-to-noise ratios shaped by graph topology and reports improved performance from the proposed methods.
- Motivation: The study examines GNN over-smoothing systematically across diverse graph datasets and models because indistinguishable representations across classes hurt performance.The authors identify limited prior explanation of why and how over-smoothing occurs.
- Metrics: MAD measures graph-representation smoothness, while MADGap measures over-smoothness by contrasting representation distances for neighboring and remote nodes.MAD is based on mean average distances among node representations; MADGap extends it using topology to estimate cross-class similarity.
- Mechanism: Smoothing is an essential property of GNNs: node-message interactions make representations closer as the number of layers increases.The observed decrease in MAD with depth supports this conclusion.
- Mechanism: Low information-to-noise ratio is identified as a key factor in over-smoothing, with graph topology influencing the balance between useful intra-class information and harmful inter-class noise.A mismatch between natural graph structure and downstream-task objectives can create excessive inter-class edges and noise during propagation.
- Methods: MADReg adds a MADGap-based training regularizer, while AdaGraph adjusts edges using model predictions to reduce over-smoothing and improve multiple GNNs.The methods respectively modify the objective and graph topology from a topological perspective.
- Evidence: MADGap is significantly correlated with model performance, supporting its use as an over-smoothing indicator.The paper reports this relationship across models and training conditions.
Datasets and Models
The experiments use node classification on seven public datasets and evaluate ten typical GNN models to assess the generality of the conclusions.
- Task: Node classification is selected because it is commonly used to evaluate GNN architectures and analyze GNN characteristics.The paper frames it as one of the most basic graph-based tasks.
- Datasets: The datasets cover citation and coauthor network types, including CORA, CiteSeer, and PubMed.The supplied passage begins listing the seven public datasets and their categories.
- Models: Ten typical GNN models are evaluated, with their propagation methods organized according to an existing propagation taxonomy.Implementation details are partly based on prior public implementations.
Topological View
The paper studies over-smoothing through graph topology, introducing MAD and MADGap to quantify representation smoothness and over-smoothness. It links over-smoothing to low information-to-noise ratios in propagated messages and shows that topology optimization can improve model performance.
- MAD: Metric for Smoothness: MAD measures graph-representation smoothness by averaging cosine distances among selected node representations.The metric uses final-layer hidden representations, computes pairwise cosine distances, masks target node pairs, and averages the resulting distances.
- MAD: Metric for Smoothness: Across ten typical GNNs, MAD decreases as layers increase, with high-layer representations approaching indistinguishability.These observations support smoothing as an essential property of GNN message propagation.
- Information-to-noise Ratio Largely Affects Over-smoothness: The information-to-noise ratio is the proportion of intra-class pairs among all node pairs that interact through message propagation, and it declines with increasing neighbor order.Higher-order interactions therefore contain relatively more inter-class noise in the examined citation datasets.
- Information-to-noise Ratio Largely Affects Over-smoothness: MADGap estimates over-smoothness as the difference between MAD for remote and neighboring nodes, with neighboring and remote sets defined by topology.The paper uses nodes at orders ≤3 for MADneb and orders ≥8 for MADrmt.
- Information-to-noise Ratio Largely Affects Over-smoothness: Higher information-to-noise ratios generally correspond to smaller over-smoothing degrees and higher prediction accuracy.The relationship is observed for node subsets under the same model and propagation depth, with similar phenomena reported on other datasets.
- Topology Affects the Information-to-noise Ratio: Graph topology affects message quality because inter-class edges introduce noise for node classification, whereas topology adjustment improves MADGap and model performance.Removing inter-class edges or adding intra-class edges produces the reported improvement, motivating topology optimization beyond architecture design.
Topological View
The paper addresses over-smoothing from a topological perspective through MADReg and AdaEdge, which respectively regularize representations and adapt graph edges. Experiments show improved performance and reduced over-smoothing, especially in deeper GNNs.
- Topological View: MADReg adds a MADGap-based regularizer to the training objective, while AdaEdge adaptively changes topology through prediction-based edge removal and addition.Both methods are designed for general GNN models rather than a specific architecture.
- MADReg: MADGap is added to training because it correlates strongly with model performance and is intended to increase useful information while reducing interference noise.The regularizer is computed on the training set alongside the cross-entropy loss.
- AdaEdge: AdaEdge first trains on the original graph, then adjusts the topology from model predictions to better match the downstream task objective.The method is motivated by adding intra-class edges and removing inter-class edges.
- Experimental Results: In four-layer GNNs on CORA, CiteSeer, and PubMed, MADReg and AdaEdge relieve severe over-smoothing and improve performance for most models.The controlled experiments cover 10 baseline GNN models, with six models shown in Figure 6 and the remainder reported in the appendix.
- Experimental Results: AdaEdge improves model performance in most cases across seven datasets, but incorrect topology adjustments can cause little or no improvement.The authors identify excessive incorrect operations as a source of interference during training.
Related Work
Prior work established GNN effectiveness and the prevalence of over-smoothing, while exploring architectures and smoothing-based remedies. This paper distinguishes reasonable smoothing from over-smoothing and identifies a lack of systematic quantitative research.
- Related Architectures: GNN research has developed attention, neighborhood-sampling, dynamic, and selective aggregation architectures for graph-based tasks.These approaches include GAT, inductive aggregation, evolving graph models, and node-adaptive locality.
- Over-smoothing Studies: Earlier studies showed that over-smoothing is common, analyzed GCN smoothing theoretically, and proposed methods for improving or overcoming smoothing-related errors.Other work also addressed over-smoothing with recurrent components in text classification.
- Over-smoothing Studies: This paper characterizes smoothing as an essential GNN feature and separates reasonable smoothing from over-smoothing using the information-to-noise ratio.The distinction links useful smoothing to GNN operation and harmful smoothing to degraded performance.
- Research Gap: Existing work commonly discusses over-smoothing as a phenomenon, but systematic and quantitative research on it remains limited.
Conclusion and Future Work
The paper concludes that quantitative analysis and topology-aware interventions can reduce over-smoothing and improve GNN performance. It also identifies incorrect graph adjustments in AdaEdge as an unresolved limitation.
- Conclusion: MAD and MADGap quantify smoothness and over-smoothness, supporting the finding that smoothing is intrinsic to GNNs and that over-mixed information and noise cause over-smoothing.The paper also reports a high correlation between MADGap and model performance.
- Conclusion: MADReg and AdaEdge effectively relieve over-smoothing and improve model performance in general situations across the reported experiments.
- Future Work: AdaEdge remains vulnerable to incorrect graph-adjustment operations, motivating future work to reduce these errors.The stated limitation concerns the adjustment process rather than the general usefulness of topology optimization.
A Experimental Settings
The experiments use seven node-classification datasets spanning citation, coauthor, and Amazon networks, together with controlled multi-run evaluation. Graphs are undirected and unweighted, and baseline and proposed methods share splits, seeds, and hyperparameters.
- Datasets: The study evaluates seven node-classification datasets from citation, coauthor, and Amazon network types.The datasets include CORA, CiteSeer, PubMed, Coauthor CS, Coauthor Physics, Amazon Computers, and Amazon Photo.
- Datasets: Citation datasets represent papers linked by citations, coauthor datasets represent authors linked by shared papers, and Amazon datasets represent goods linked by co-purchases.
- Datasets: All experimental graphs are undirected and have no edge weights.
- Evaluation Protocol: The evaluation uses repeated dataset splits and random seeds because split choice can substantially affect semi-supervised node-classification results.The reported Figure 6 statistics aggregate 50 runs formed from five splitting methods and ten seeds per split.
- Evaluation Protocol: Controlled comparisons reuse the same split and seed lists for baseline and proposed methods and fix the other hyperparameters.This design reduces random variation when comparing methods.
- Notation: For node classification, the graph is represented by an adjacency matrix and node-feature matrix, with labels used to predict node classes.
B MAD Global Values on More Datasets
Across CiteSeer and PubMed, MAD decreases as GNN depth increases, indicating progressively smoother node representations.
- MAD values decrease as the number of GNN layers increases on both CiteSeer and PubMed.
- The declining MAD trend occurs across all evaluated baseline GNN models.
- The result provides a cross-dataset observation of increasing representation smoothness with model depth.
C Infomation-to-noise Ratio Experiment on More Datasets
The information-to-noise ratio strongly affects over-smoothness: higher intra-class node ratios increase both model performance and MADGap. The experiment evaluates this relationship on CiteSeer and PubMed across four GNN models.
- Model performance and MADGap increase as the intra-class node ratio rises in four GNN models.The experiment uses 2-order neighbours on CiteSeer and PubMed.
- A higher intra-class node ratio provides more useful information for graph nodes.
- Information-to-noise ratio largely affects node-representation over-smoothness.
E Supplementary Result of Relieving Over-smoothing
Additional experiments examine high-layer GNNs, where over-smoothing is serious, and report that the proposed methods relieve over-smoothing while improving performance in most cases.
- Figure 8 reports results for additional GNN models at four layers, where over-smoothing is serious.
- The additional experiments evaluate methods for relieving over-smoothing in high-layer GNNs.
- The reported results concern the relationship between over-smoothing relief and model performance.
F Error Analysis of MADReg
MADReg is most effective for deeper GCNs, where over-smoothing is more severe, while shallow models show limited performance gains. AdaEdge adjusts graph topology using prediction-based edge operations and iterative evaluation.
- MADReg relieves over-smoothing and improves GCN performance especially as the number of layers increases.The comparison covers CORA, CiteSeer, and PubMed.
- Shallow GCNs show limited performance improvement from MADReg because over-smoothing is not serious after training.
- The topology adjustment can add edges between same-prediction, high-confidence nodes and remove edges between different-prediction, high-confidence nodes.
- Edge addition and removal can be ordered in either sequence through the ADJUSTGRAPH procedure.
- AdaEdge iteratively adjusts graph topology using GNN predictions, confidences, and accuracy-based stopping.