  Given a vertex of interest in a network $G_1$, the vertex nomination problem
seeks to find the corresponding vertex of interest (if it exists) in a second
network $G_2$. A vertex nomination scheme produces a list of the vertices in
$G_2$, ranked according to how likely they are judged to be the corresponding
vertex of interest in $G_2$. The vertex nomination problem and related
information retrieval tasks have attracted much attention in the machine
learning literature, with numerous applications to social and biological
networks. However, the current framework has often been confined to a
comparatively small class of network models, and the concept of statistically
consistent vertex nomination schemes has been only shallowly explored. In this
paper, we extend the vertex nomination problem to a very general statistical
model of graphs. Further, drawing inspiration from the long-established
classification framework in the pattern recognition literature, we provide
definitions for the key notions of Bayes optimality and consistency in our
extended vertex nomination framework, including a derivation of the Bayes
optimal vertex nomination scheme. In addition, we prove that no universally
consistent vertex nomination schemes exist. Illustrative examples are provided
throughout.
