Inference for multiple heterogeneous networks with a common invariant subspace

The development of models and methodology for the analysis of data from multiple heterogeneous networks is of importance both in statistical network theory and across a wide spectrum of appli- cation domains. Although single-graph analysis 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 multiple random graph model, which describes a heterogeneous collection of networks with a shared la- tent structure on the vertices but potentially different connectivity patterns for each graph. The model encompasses many popular network representations, including the stochastic blockmodel. The model is both exible enough to meaningfully account for important graph differences, and tractable enough to allow for accurate inference in multiple networks. In particular, a joint spec- tral embedding of adjacency matrices|the multiple adjacency spectral embedding|leads to si- multaneous consistent estimation of underlying parameters for each graph. Under mild additional assumptions, the estimates satisfy asymptotic normality and yield improvements for graph eigen- value estimation. In both simulated and real data, the model and the embedding can be deployed for a number of subsequent network inference tasks, including dimensionality reduction, classifica- tion, hypothesis testing, and community detection. Specifically, when the embedding is applied to a data set of connectomes constructed through diffusion magnetic resonance imaging, the result is an accurate classification of brain scans by human subject and a meaningful determination of heterogeneity across scans of different individuals.