  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 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. For exact
classification, we show that the separability threshold can be improved
exponentially up to $O({\log{n}}/{\log\log{n}})$ corrected convolutions.
