  Performing statistical analyses on collections of graphs is of import to many
disciplines, but principled, scalable methods for multi-sample graph inference
are few. Here we describe an "omnibus" embedding in which multiple graphs on
the same vertex set are jointly embedded into a single space with a distinct
representation for each graph. We prove a central limit theorem for this
embedding and demonstrate how it streamlines graph comparison, obviating the
need for pairwise subspace alignments. The omnibus embedding achieves
near-optimal inference accuracy when graphs arise from a common distribution
and yet retains discriminatory power as a test procedure for the comparison of
different graphs. Moreover, this joint embedding and the accompanying central
limit theorem are important for answering multiscale graph inference questions,
such as the identification of specific subgraphs or vertices responsible for
similarity or difference across networks. We illustrate this with a pair of
analyses of connectome data derived from dMRI and fMRI scans of human subjects.
In particular, we show that this embedding allows the identification of
specific brain regions associated with population-level differences. Finally,
we sketch how the omnibus embedding can be used to address pressing open
problems, both theoretical and practical, in multisample graph inference.

Methodology