  Graph-based semi-supervised regression (SSR) is the problem of estimating the
value of a function on a weighted graph from its values (labels) on a small
subset of the vertices. This paper is concerned with the consistency of SSR in
the context of classification, in the setting where the labels have small noise
and the underlying graph weighting is consistent with well-clustered nodes. We
present a Bayesian formulation of SSR in which the weighted graph defines a
Gaussian prior, using a graph Laplacian, and the labeled data defines a
likelihood. We analyze the rate of contraction of the posterior measure around
the ground truth in terms of parameters that quantify the small label error and
inherent clustering in the graph. We obtain bounds on the rates of contraction
and illustrate their sharpness through numerical experiments. The analysis also
gives insight into the choice of hyperparameters that enter the definition of
the prior.
