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In this paper, based on a weighted projection of a bipartite user-object network, we introduce a personalized recommendation algorithm, called network-based inference (NBI), which has higher accuracy than the classical algorithm, namely collaborative filtering. In NBI, the correlation resulting from a specific attribute may be repeatedly counted in the cumulative recommendations from different objects. By considering the higher order correlations, we design an improved algorithm that can, to some extent, eliminate the redundant correlations. We test our algorithm on two benchmark data sets, MovieLens and Netflix. Compared with NBI, the algorithmic accuracy, measured by the ranking score, can be further improved by 23 per cent for MovieLens and 22 per cent for Netflix. The present algorithm can even outperform the Latent Dirichlet Allocation algorithm, which requires much longer computational time …