  Complex interactions between entities are often represented as edges in a
network. In practice, the network is often constructed from noisy measurements
and inevitably contains some errors. In this paper we consider the problem of
estimating a network from multiple noisy observations where edges of the
original network are recorded with both false positives and false negatives.
This problem is motivated by neuroimaging applications where brain networks of
a group of patients with a particular brain condition could be viewed as noisy
versions of an unobserved true network corresponding to the disease. The key to
optimally leveraging these multiple observations is to take advantage of
network structure, and here we focus on the case where the true network
contains communities. Communities are common in real networks in general and in
particular are believed to be presented in brain networks. Under a community
structure assumption on the truth, we derive an efficient method to estimate
the noise levels and the original network, with theoretical guarantees on the
convergence of our estimates. We show on synthetic networks that the
performance of our method is close to an oracle method using the true parameter
values, and apply our method to fMRI brain data, demonstrating that it
constructs stable and plausible estimates of the population network.
