Inference for multiple heterogeneous networks with a common invariant\n subspace

Abstract

The development of models for multiple heterogeneous network data is of\ncritical importance both in statistical network theory and across multiple\napplication domains. Although single-graph inference is well-studied, multiple\ngraph inference is largely unexplored, in part because of the challenges\ninherent in appropriately modeling graph differences and yet retaining\nsufficient model simplicity to render estimation feasible. This paper addresses\nexactly this gap, by introducing a new model, the common subspace\nindependent-edge (COSIE) multiple random graph model, which describes a\nheterogeneous collection of networks with a shared latent structure on the\nvertices but potentially different connectivity patterns for each graph. The\nCOSIE model encompasses many popular network representations, including the\nstochastic blockmodel. The model is both flexible enough to meaningfully\naccount for important graph differences and tractable enough to allow for\naccurate inference in multiple networks. In particular, a joint spectral\nembedding of adjacency matrices - the multiple adjacency spectral embedding\n(MASE) - leads, in a COSIE model, to simultaneous consistent estimation of\nunderlying parameters for each graph. Under mild additional assumptions, MASE\nestimates satisfy asymptotic normality and yield improvements for graph\neigenvalue estimation and hypothesis testing. In both simulated and real data,\nthe COSIE model and the MASE embedding can be deployed for a number of\nsubsequent network inference tasks, including dimensionality reduction,\nclassification, hypothesis testing and community detection. Specifically, when\nMASE is applied to a dataset of connectomes constructed through diffusion\nmagnetic resonance imaging, the result is an accurate classification of brain\nscans by patient and a meaningful determination of heterogeneity across scans\nof different subjects.\n