  We theoretically study semi-supervised clustering in sparse graphs in the
presence of pairwise constraints on the cluster assignments of nodes. We focus
on bi-cluster graphs, and study the impact of semi-supervision for varying
constraint density and overlap between the clusters. Recent results for
unsupervised clustering in sparse graphs indicate that there is a critical
ratio of within-cluster and between-cluster connectivities below which clusters
cannot be recovered with better than random accuracy. The goal of this paper is
to examine the impact of pairwise constraints on the clustering accuracy. Our
results suggests that the addition of constraints does not provide automatic
improvement over the unsupervised case. When the density of the constraints is
sufficiently small, their only impact is to shift the detection threshold while
preserving the criticality. Conversely, if the density of (hard) constraints is
above the percolation threshold, the criticality is suppressed and the
detection threshold disappears.
