  We analyze graph smoothing with \emph{mean aggregation}, where each node
successively receives the average of the features of its neighbors. Indeed, it
has quickly been observed that Graph Neural Networks (GNNs), which generally
follow some variant of Message-Passing (MP) with repeated aggregation, may be
subject to the oversmoothing phenomenon: by performing too many rounds of MP,
the node features tend to converge to a non-informative limit. In the case of
mean aggregation, for connected graphs, the node features become constant
across the whole graph. At the other end of the spectrum, it is intuitively
obvious that some MP rounds are necessary, but existing analyses do not exhibit
both phenomena at once: beneficial ``finite'' smoothing and oversmoothing in
the limit. In this paper, we consider simplified linear GNNs, and rigorously
analyze two examples for which a finite number of mean aggregation steps
provably improves the learning performance, before oversmoothing kicks in. We
consider a latent space random graph model, where node features are partial
observations of the latent variables and the graph contains pairwise
relationships between them. We show that graph smoothing restores some of the
lost information, up to a certain point, by two phenomenon: graph smoothing
shrinks non-principal directions in the data faster than principal ones, which
is useful for regression, and shrinks nodes within communities faster than they
collapse together, which improves classification.
