Figure 1. CAR index.

When a link or a node directly interacts with a seed node, it belongs to the first-level-neighbourhood and conveys first-level topological information. Conversely, a link or a node that interacts with the first-level-neighbourhood conveys second-level information. Second-level information is valuable and its use can significantly improve topological link-prediction, but unfortunately it is also very noisy, and for this reason difficult to integrate with the first-level information. CAR is designed to capture and filter meaningful second-level information by exploiting common-first-neighbours. The topological information conveyed by the internal links between common-first-neighbours is valuable second-level information. Indeed, the more the common-first-neighbours reciprocally interact, the more they represent a local-community, which in turn encompasses the two seed nodes and thus increases their candidate-link likelihood. Here we introduce the idea that the likelihood of a candidate-link is a function of both the number of common-first-neighbours (as in the CN index) and of the number of links between them (local-community links), as expressed in the formula of CAR. The two explicative examples clarify how CAR increases discriminative resolution between candidate-links with the same number of common-first-neighbours.