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Analysis of Corrected Graph Convolutions

Robert Wang, Aseem Baranwal, Kimon Fountoulakis

TL;DR

This work addresses oversmoothing in graph convolutional networks by introducing corrected adjacency operators that remove the principal eigenvector, shifting the limiting behavior from the top to the second eigenvector. Using a spectral analysis within the contextual stochastic block model, it proves that each corrected convolution round reduces misclassification error exponentially up to a saturation point for binary classification, and exponentially improves the exact separability threshold to $O\left(\frac{\log n}{\log\log n}\right)$ under suitable density and signal strength. The results extend from two-class CSBM to multi-class Gaussian mixtures, showing that a softmax on contracted features yields accurate classification when centers are sufficiently separated relative to graph noise. The paper also provides concentration bounds, reductions to 1-D analysis, and empirical validation on synthetic and real graphs, illustrating practical benefits for mitigating oversmoothing in GNNs. Overall, it offers rigorous, scalable guarantees for using corrected graph convolutions to improve node classification performance in graphs with realistic block-structured signals.

Abstract

Machine learning for node classification on graphs is a prominent area driven by applications such as recommendation systems. State-of-the-art models often use multiple graph convolutions on the data, as empirical evidence suggests they can enhance performance. However, it has been shown empirically and theoretically, that too many graph convolutions can degrade performance significantly, a phenomenon known as oversmoothing. In this paper, we provide a rigorous theoretical analysis, based on the two-class contextual stochastic block model (CSBM), of the performance of vanilla graph convolution from which we remove the principal eigenvector to avoid oversmoothing. We perform a spectral analysis for $k$ rounds of corrected graph convolutions, and we provide results for partial and exact classification. For partial classification, we show that each round of convolution can reduce the misclassification error exponentially up to a saturation level, after which performance does not worsen. We also extend this analysis to the multi-class setting with features distributed according to a Gaussian mixture model. For exact classification, we show that the separability threshold can be improved exponentially up to $O({\log{n}}/{\log\log{n}})$ corrected convolutions.

Analysis of Corrected Graph Convolutions

TL;DR

This work addresses oversmoothing in graph convolutional networks by introducing corrected adjacency operators that remove the principal eigenvector, shifting the limiting behavior from the top to the second eigenvector. Using a spectral analysis within the contextual stochastic block model, it proves that each corrected convolution round reduces misclassification error exponentially up to a saturation point for binary classification, and exponentially improves the exact separability threshold to under suitable density and signal strength. The results extend from two-class CSBM to multi-class Gaussian mixtures, showing that a softmax on contracted features yields accurate classification when centers are sufficiently separated relative to graph noise. The paper also provides concentration bounds, reductions to 1-D analysis, and empirical validation on synthetic and real graphs, illustrating practical benefits for mitigating oversmoothing in GNNs. Overall, it offers rigorous, scalable guarantees for using corrected graph convolutions to improve node classification performance in graphs with realistic block-structured signals.

Abstract

Machine learning for node classification on graphs is a prominent area driven by applications such as recommendation systems. State-of-the-art models often use multiple graph convolutions on the data, as empirical evidence suggests they can enhance performance. However, it has been shown empirically and theoretically, that too many graph convolutions can degrade performance significantly, a phenomenon known as oversmoothing. In this paper, we provide a rigorous theoretical analysis, based on the two-class contextual stochastic block model (CSBM), of the performance of vanilla graph convolution from which we remove the principal eigenvector to avoid oversmoothing. We perform a spectral analysis for rounds of corrected graph convolutions, and we provide results for partial and exact classification. For partial classification, we show that each round of convolution can reduce the misclassification error exponentially up to a saturation level, after which performance does not worsen. We also extend this analysis to the multi-class setting with features distributed according to a Gaussian mixture model. For exact classification, we show that the separability threshold can be improved exponentially up to corrected convolutions.
Paper Structure (28 sections, 26 theorems, 85 equations, 4 figures)

This paper contains 28 sections, 26 theorems, 85 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Literature review
  4. Preliminaries and Model Description
  5. Contextual Stochastic Block Model
  6. Results and Interpretation
  7. Discussion on our Assumptions
  8. One-Dimensional CSBM
  9. Partial Classification
  10. Exact Classification
  11. Multi-class Analysis on Gaussian Mixture Model
  12. Experiments
  13. Synthetic Data
  14. Real Data
  15. Conclusion and Future Work
  16. ...and 13 more sections

Key Result

Theorem 4.1

Suppose we are given a 2-block $m$-dimensional CSBM with parameters $n,p > q, \mu, \nu, \sigma$ satisfying $\gamma(p,q) :=\frac{p-q}{p+q} \geq \Omega\left(\sqrt{\frac{1}{np}}\right)$ and $p\geq \Omega\left(\frac{\log^2{n}}{n}\right)$. There exists a linear classifier such that after $k$ rounds of co vertices, where $C$ is an absolute constant. Furthermore, if $\gamma \geq \Omega(\sqrt{\frac{\log{n

Figures (4)

  • Figure 1: Accuracy plot (average over 50 trials) against the signal-to-noise ratio of the features (ratio of the distance between the means to the standard deviation) for increasing number of convolutions. Here, $v = D^{1/2}\mathbbm{1}$ and the "GCN with $vv^\top$ removed" refers to convolution with the corrected, normalized adjacency matrix. "GCN with $\mathbbm{1}\mathbbm{1}^\top$ removed" is the corrected, unnormalized matrix.
  • Figure 2: Accuracy plot (average over 50 trials) against graph relative signal strength ($\gamma=|p-q|/(p+q)$) for various values of the number of convolutions.
  • Figure 3: Accuracy plots (average over 50 trials) against the number of layers for real datasets.
  • Figure 4: Accuracy plot (averaged over 50 trials) on CSBM data with $5$ balanced classes, $500$ nodes per class and orthogonal means, with fixed $p=0.1$.

Theorems & Definitions (43)

  • Theorem 4.1
  • Theorem 4.2
  • Lemma 5.1
  • Proposition 5.2
  • Proposition 6.1
  • Proposition 7.1
  • Theorem 8.1
  • Theorem A.1
  • Lemma A.2
  • proof
  • ...and 33 more