Network-structured data arise in many applications; joint analysis of multiple networks is especially relevant to applications like neuroimaging, where each network corresponds to a patient's brain connectome. Models for multiple networks have typically focused on estimating their shared structure. Two-sample tests have also been developed, testing some version of the hypothesis of two samples of networks being indistinguishable across all node pairs (the global null). However, scientifically relevant hypotheses rarely take this form: for example, in neuroimaging it is rarely of interest to compare whole brains of patients and healthy controls, and the focus is more often on a particular brain region. Beyond comparisons, it is also of interest to estimate structures that correspond to a specific subgroup of networks, for example, for patients with a certain trait. One could always do that using just the subgroup of interest, but using all available samples allows us to better estimate structures that are shared by all, which in turn helps separate out the structure associated with a trait. This talk will introduce two methods that help address these challenges: mesoscale testing on networks, which conducts formal hypothesis testing on a subset of edges (like a brain region) while leveraging the rest of the network to increase power; and group MultiNeSS, a method that takes a sample of networks and estimates structures that are shared by all, specific to groups, or just unique to an individual network. In both cases, we leverage the assumption of low-rank expectation of adjacency matrices which has been observed widely in practice. Based on joint work with Peter MacDonald, Alexander Kagan, and Ji Zhu.
