example, well known in decision theory and preference modeling that transitive preference relations result in utility functions [@Luce1965; @Bodenhofer2007]. Learning reciprocal or antisymmetric relations can be interpreted as learning edges in a directed graph.

Ordinary binary relations. Many applications can be found where neither symmetry nor reciprocity holds. From a graph inference perspective, learning such relations should be seen as learning the edges in a bidirectional graph, where edges in one direction do not impose constraints on edges in the other direction.

Indeed, the framework that we propose below strongly relies on graphs, where nodes represent the data objects that are studied and the edges represent the relations present in the training set. The weights on the edges characterize the values of known relations, while unconnected nodes indicate pairs of objects for which the unknown relation needs to be predicted. The left graph in Figure \[fig:examples\] visualizes a toy example representing the most general case where neither symmetry nor reciprocity holds. Depending on the application, the learning algorithm should try to predict the relations for three types of object pairs:

-   pairs of objects that are already present in the training dataset by means of other edges, like the pair 