Fig. 1

Network similarity does not imply connectivity. a In social networks, a large number of common friends implies a higher chance to become friends (red link between nodes X and Y), known as the Triadic Closure Principle (TCP). TCP predicts (P) links based on node similarity (S), quantifying the number of shared neighbors between each node pair (A²). b A basic mathematical formulation of TCP implies that protein pairs of high Jaccard similarity are more likely to interact. c We do not observe the expected trend in Protein-Protein Interaction (PPI) datasets, as illustrated here for a binary human PPI network (HI-II-14)⁵: high Jaccard similarity indicates a lower chance for the proteins to interact (see Supplementary Fig. 3 for further networks). The data are binned logarithmically based on the Jaccard similarity values. d PPIs often require complementary interfaces^10,11, hence, two proteins, X and Y, with similar interfaces share many of their neighbors. Yet, a shared interface does not typically guarantee that X and Y directly interact with each other (see Supplementary Fig. 1 for an illustration with known 3D structures). Instead, an additional interaction partner of X (protein D) might be also shared with protein Y (blue link). Such a link can be predicted by using paths of length 3 (L3). L3 identifies similar nodes to the known partners (P = AS), going one step beyond the similarity-based argument of TCP. e Even without using any structural information, two proteins, such as Y and D are expected to interact if they are linked by multiple \(\ell = 3\) paths in the network (L3). f As opposed to c, we observe a strong positive trend in HI-II-14 between the probability of two proteins interacting and the number of \(\ell = 3\) paths between them, supporting the validity of the L3 principle