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The estimation of probabilities of network edges from the observed adjacency matrix has important applications to the prediction of missing links and to network denoising. It is usually addressed by estimating the graphon, a function that determines the matrix of edge probabilities, but this is ill-defined without strong assumptions on the network structure. Here we propose a novel computationally efficient method, based on neighbourhood smoothing, to estimate the expectation of the adjacency matrix directly, without making the structural assumptions that graphon estimation requires. The neighbourhood smoothing method requires little tuning, has a competitive mean squared error rate and outperforms many benchmark methods for link prediction in simulated and real networks.