A central limit theorem for an omnibus embedding of multiple random graphs and implications for multiscale network inference

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.