Despite exceptional experimental efforts to map out the human interactome, the continued data incompleteness limits our ability to understand the molecular roots of human disease. Computational tools offer a promising alternative, helping identify biologically significant, yet unmapped protein-protein interactions (PPIs). While link prediction methods connect proteins on the basis of biological or network-based similarity, interacting proteins are not necessarily similar and similar proteins do not necessarily interact. Here, we offer structural and evolutionary evidence that proteins interact not if they are similar to each other, but if one of them is similar to the other's partners. This approach, that mathematically relies on network paths of length three (L3), significantly outperforms all existing link prediction methods. Given its high accuracy, we show that L3 can offer mechanistic insights into disease mechanisms and can complement future experimental efforts to complete the human interactome.