  Spectral inference on multiple networks is a rapidly-developing subfield of
graph statistics. Recent work has demonstrated that joint, or simultaneous,
spectral embedding of multiple independent networks can deliver more accurate
estimation than individual spectral decompositions of those same networks. Such
inference procedures typically rely heavily on independence assumptions across
the multiple network realizations, and even in this case, little attention has
been paid to the induced network correlation in such joint embeddings. Here, we
present a generalized omnibus embedding methodology and provide a detailed
analysis of this embedding across both independent and correlated networks, the
latter of which significantly extends the reach of such procedures. We describe
how this omnibus embedding can itself induce correlation, leading us to
distinguish between inherent correlation -- the correlation that arises
naturally in multisample network data -- and induced correlation, which is an
artifice of the joint embedding methodology. We show that the generalized
omnibus embedding procedure is flexible and robust, and prove both consistency
and a central limit theorem for the embedded points. We examine how induced and
inherent correlation can impact inference for network time series data, and we
provide network analogues of classical questions such as the effective sample
size for more generally correlated data. Further, we show how an appropriately
calibrated generalized omnibus embedding can detect changes in real biological
networks that previous embedding procedures could not discern, confirming that
the effect of inherent and induced correlation can be subtle and
transformative, with import in theory and practice.
