  Given a network and a subset of interesting vertices whose identities are
only partially known, the vertex nomination problem seeks to rank the remaining
vertices in such a way that the interesting vertices are ranked at the top of
the list. An important variant of this problem is vertex nomination in the
multi-graphs setting. Given two graphs $G_1, G_2$ with common vertices and a
vertex of interest $x \in G_1$, we wish to rank the vertices of $G_2$ such that
the vertices most similar to $x$ are ranked at the top of the list. The current
paper addresses this problem and proposes a method that first applies adjacency
spectral graph embedding to embed the graphs into a common Euclidean space, and
then solves a penalized linear assignment problem to obtain the nomination
lists. Since the spectral embedding of the graphs are only unique up to
orthogonal transformations, we present two approaches to eliminate this
potential non-identifiability. One approach is based on orthogonal Procrustes
and is applicable when there are enough vertices with known correspondence
between the two graphs. Another approach uses adaptive point set registration
and is applicable when there are few or no vertices with known correspondence.
We show that our nomination scheme leads to accurate nomination under a
generative model for pairs of random graphs that are approximately low-rank and
possibly with pairwise edge correlations. We illustrate our algorithm's
performance through simulation studies on synthetic data as well as analysis of
a high-school friendship network and analysis of transition rates between web
pages on the Bing search engine.
