Graph edges, along with their labels, can represent information of fundamental importance, such as links between web pages, friendship between users, the rating given by users to other users or items, and much more. We introduce LEAP, a trainable, general framework for predicting the presence and properties of edges on the basis of the local structure, topology, and labels of the graph. The LEAP framework is based on the exploration and machine-learning aggregation of the paths connecting nodes in a graph. We provide several methods for performing the aggregation phase by training path aggregators, and we demonstrate the flexibility and generality of the framework by applying it to the prediction of links and user ratings in social networks. We validate the LEAP framework on two problems: link prediction, and user rating prediction. On eight large datasets, among which the arXiv collaboration network, the Yeast protein-protein interaction, and the US airlines routes network, we show that the link prediction performance of LEAP is at least as good as the current state of the art methods, such as SEAL and WLNM. Next, we consider the problem of predicting user ratings on other users: this problem is known as the edge-weight prediction problem in weighted signed networks (WSN). On Bitcoin networks, and Wikipedia RfA, we show that LEAP performs consistently better than the Fairness & Goodness based regression models, varying the amount of training edges between 10 to 90%. These examples demonstrate that LEAP, in spite of its generality, can match or best the performance of approaches that have been especially crafted to solve very specific edge prediction problems.