  Statistical inference on graphs is a burgeoning field in the applied and
theoretical statistics communities, as well as throughout the wider world of
science, engineering, business, etc. In many applications, we are faced with
the reality of errorfully observed graphs. That is, the existence of an edge
between two vertices is based on some imperfect assessment. In this paper, we
consider a graph $G = (V,E)$. We wish to perform an inference task -- the
inference task considered here is "vertex classification". However, we do not
observe $G$; rather, for each potential edge $uv \in {{V}\choose{2}}$ we
observe an "edge-feature" which we use to classify $uv$ as edge/not-edge. Thus
we errorfully observe $G$ when we observe the graph $\widetilde{G} =
(V,\widetilde{E})$ as the edges in $\widetilde{E}$ arise from the
classifications of the "edge-features", and are expected to be errorful.
Moreover, we face a quantity/quality trade-off regarding the edge-features we
observe -- more informative edge-features are more expensive, and hence the
number of potential edges that can be assessed decreases with the quality of
the edge-features. We studied this problem by formulating a quantity/quality
tradeoff for a simple class of random graphs model, namely the stochastic
blockmodel. We then consider a simple but optimal vertex classifier for
classifying $v$ and we derive the optimal quantity/quality operating point for
subsequent graph inference in the face of this trade-off. The optimal operating
points for the quantity/quality trade-off are surprising and illustrate the
issue that methods for intermediate tasks should be chosen to maximize
performance for the ultimate inference task. Finally, we investigate the
quantity/quality tradeoff for errorful obesrvations of the {\it C.\ elegans}
connectome graph.
