  In increasingly many settings, data sets consist of multiple samples from a
population of networks, with vertices aligned across these networks. For
example, brain connectivity networks in neuroscience consist of measures of
interaction between brain regions that have been aligned to a common template.
We consider the setting where the observed networks have a shared expectation,
but may differ in the noise structure on their edges. Our approach exploits the
shared mean structure to denoise edge-level measurements of the observed
networks and estimate the underlying population-level parameters. We also
explore the extent to which edge-level errors influence estimation and
downstream inference. We establish a finite-sample concentration inequality for
the low-rank eigenvalue truncation of a random weighted adjacency matrix that
may be of independent interest. The proposed approach is illustrated on
synthetic networks and on data from an fMRI study of schizophrenia.
