A central limit theorem for an omnibus embedding of random dot product graphs

Performing statistical analyses on collections of graphs is of import to many disciplines, but principled, scalable methods for multisample graph inference are few. In this paper, 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 omnibus embedding, and we show that this simultaneous embedding into a single common space allows for the comparison of graphs without the requirement that the embedded points associated to each graph undergo cumbersome pairwise alignments. Moreover, the existence of multiple embedded points for each vertex renders possible the resolution of important multiscale graph inference goals, such as the identification of specific subgraphs or vertices as drivers of similarity or difference across large networks. 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. We demonstrate the applicability of the omnibus embedding in two analyses of connectomic graphs generated from MRI scans of the brain in human subjects. We show how the omnibus embedding can be used to detect statistically significant differences, at multiple scales, across these networks, with an identification of specific brain regions that are associated with these 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.