  Motivated by multi-subject experiments in neuroimaging studies, we develop a
modeling framework for joint community detection in a group of related
networks, which can be considered as a sample from a population of networks.
The proposed random effects stochastic block model facilitates the study of
group differences and subject-specific variations in the community structure.
The model proposes a putative mean community structure which is representative
of the group or the population under consideration but is not the community
structure of any individual component network. Instead, the community
memberships of nodes vary in each component network with a transition matrix,
thus modeling the variation in community structure across a group of subjects.
To estimate the quantities of interest we propose two methods, a variational EM
algorithm, and a model-free "two-step" method based on either spectral or
non-negative matrix factorization (NMF). Our NMF based method Co-OSNTF is of
independent interest and we study its convergence properties to a stationary
point. We also develop a resampling-based hypothesis test for differences in
community structure in two populations both at the whole network level and node
level. The methodology is applied to a publicly available fMRI dataset from
multi-subject experiments involving schizophrenia patients. Our methods reveal
an overall putative community structure representative of the group as well as
subject-specific variations within each group. Using our network level
hypothesis tests we are able to ascertain statistically significant difference
in community structure between the two groups, while our node level tests help
determine the nodes that are driving the difference.
