Networks are often recorded with measurement errors, and therefore the observed links may not always correspond to the true relations between nodes. Addressing this problem requires predicting potentially missing links and assessing the strength of observed links, which can be done by estimating the expectation of the network adjacency matrix. We propose a low-rank effects model for the expected adjacency matrix, combining generalized linear models and matrix completion techniques. This model can be applied to various types of networks including directed, undirected, binary, and weighted networks, and can utilize additional information on node and edge covariates. We provide a consistent maximum likelihood estimator with an error bound under additional conditions. The estimate can be calculated efficiently via a projected gradient ascent algorithm. The tuning parameter selection can be done by cross-validation, or the computational cost can be further reduced by using the BIC criterion. The method gives promising empirical results on both simulated data and real networks, including a Facebook network and a neural network.