Analysis of Corrected Graph Convolutions

TL;DR

This paper provides a rigorous spectral analysis of corrected graph convolutions, demonstrating their effectiveness in improving node classification performance and mitigating oversmoothing in graph neural networks.

Contribution

It introduces a theoretical framework analyzing corrected graph convolutions using stochastic block models, showing exponential error reduction and improved separability thresholds.

Findings

Each convolution reduces misclassification error exponentially in partial classification.

Corrected convolutions extend the separability threshold exponentially in the multi-class setting.

Performance saturates after a certain number of convolution rounds, preventing degradation.

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.