We introduce an efficient algorithm for reducing bond dimensions in an arbitrary tensor network without changing its geometry. The method is based on a quantitative understanding of local correlations in a network. Together with a tensor network coarse-graining algorithm, it yields a proper renormalization group (RG) flow. Compared to existing methods, the advantages of our algorithm are its low computational cost, simplicity of implementation, and applicability to any network. We benchmark it by evaluating physical observables for the two-dimensional classical Ising model and find accuracy comparable with the best existing tensor network methods. Because of its graph independence, our algorithm is an excellent candidate for implementation of real-space RG in higher dimensions. We discuss some of the details and the remaining challenges in three dimensions. Source code for our algorithm is freely available.