  The development of models for multiple heterogeneous network data is of
critical importance both in statistical network theory and across multiple
application domains. Although single-graph inference is well-studied, multiple
graph inference is largely unexplored, in part because of the challenges
inherent in appropriately modeling graph differences and yet retaining
sufficient model simplicity to render estimation feasible. This paper addresses
exactly this gap, by introducing a new model, the common subspace
independent-edge (COSIE) multiple random graph model, which describes a
heterogeneous collection of networks with a shared latent structure on the
vertices but potentially different connectivity patterns for each graph. The
COSIE model encompasses many popular network representations, including the
stochastic blockmodel. The model is both flexible enough to meaningfully
account for important graph differences and tractable enough to allow for
accurate inference in multiple networks. In particular, a joint spectral
embedding of adjacency matrices - the multiple adjacency spectral embedding
(MASE) - leads, in a COSIE model, to simultaneous consistent estimation of
underlying parameters for each graph. Under mild additional assumptions, MASE
estimates satisfy asymptotic normality and yield improvements for graph
eigenvalue estimation and hypothesis testing. In both simulated and real data,
the COSIE model and the MASE embedding can be deployed for a number of
subsequent network inference tasks, including dimensionality reduction,
classification, hypothesis testing and community detection. Specifically, when
MASE is applied to a dataset of connectomes constructed through diffusion
magnetic resonance imaging, the result is an accurate classification of brain
scans by patient and a meaningful determination of heterogeneity across scans
of different subjects.
