Recovering low-rank structure from multiple networks with unknown edge distributions

In many applications, simultaneous analysis of multiple networks is of increasing interest. Broadly speaking, a researcher may be interested in identifying structure that is shared across multiple networks. In the social sciences, this may correspond to some common underlying structure that appears, for example, in different friendship networks across high schools. In biology, one may be interested in identifying the extent to which different organisms' protein-protein interaction networks display a similar structure. In neuroscience, one may wish to identify common patterns across multiple subjects' brains in an imaging study. This last application in particular is easily abstracted to the situation where one observes a collection of independent graphs on the same vertex set, as there are well-established and widely used algorithms that map locations in individual brains onto an atlas of so-called regions of interest (ROIs), such as the one developed by Power et al. (2011). The assumption of vertex alignment across graphs is common in the statistics literature on multiple network analysis for neuroimaging applications; see for example Levin et al. (2017); Arroyo-Relión et al. (2017).