  Community structure in networks is observed in many different domains, and
unsupervised community detection has received a lot of attention in the
literature. Increasingly the focus of network analysis is shifting towards
using network information in some other prediction or inference task rather
than just analyzing the network itself. In particular, in neuroimaging
applications brain networks are available for multiple subjects and the goal is
often to predict a phenotype of interest. Community structure is well known to
be a feature of brain networks, typically corresponding to different regions of
the brain responsible for different functions. There are standard parcellations
of the brain into such regions, usually obtained by applying clustering methods
to brain connectomes of healthy subjects. However, when the goal is predicting
a phenotype or distinguishing between different conditions, these static
communities from an unrelated set of healthy subjects may not be the most
useful for prediction. Here we present a method for supervised community
detection, aiming to find a partition of the network into communities that is
most useful for predicting a particular response. We use a block-structured
regularization penalty combined with a prediction loss function, and compute
the solution with a combination of a spectral method and an ADMM optimization
algorithm. We show that the spectral clustering method recovers the correct
communities under a weighted stochastic block model. The method performs well
on both simulated and real brain networks, providing support for the idea of
task-dependent brain regions.
