  Outliers arise in networks due to different reasons such as fraudulent
behavior of malicious users or default in measurement instruments and can
significantly impair network analyses. In addition, real-life networks are
likely to be incompletely observed, with missing links due to individual
non-response or machine failures. Identifying outliers in the presence of
missing links is therefore a crucial problem in network analysis. In this work,
we introduce a new algorithm to detect outliers in a network that
simultaneously predicts the missing links. The proposed method is statistically
sound: we prove that, under fairly general assumptions, our algorithm exactly
detects the outliers, and achieves the best known error for the prediction of
missing links with polynomial computation cost. It is also computationally
efficient: we prove sub-linear convergence of our algorithm. We provide a
simulation study which demonstrates the good behavior of the algorithm in terms
of outliers detection and prediction of the missing links. We also illustrate
the method with an application in epidemiology, and with the analysis of a
political Twitter network. The method is freely available as an R package on
the Comprehensive R Archive Network.
