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.