the Kronecker product pairwise kernel. By construction, the Cartesian pairwise kernel has important limitations, since it cannot generalize to couples of nodes for which both nodes did not appear in the training dataset.

Special relations
=================

Thus, if no further information is available about the relation that underlies the data, one should definitely use the Kronecker product edge kernel. In this most general case, we allow that for any pair of nodes in the graph several edges can exist, in which an edge in one direction does not necessarily impose constraints on the edge in the opposite direction. Multiple edges in the same direction can connect two nodes, leading to a multi-graph as in Figure \[fig:examples\], where two different edges in the same direction connect nodes $D$ and $E$. This construction is required to allow repeated measurements. However, two important subclasses of relations deserve further attention: reciprocal relations and symmetric relations.

Reciprocal relations
--------------------

This subsection briefly summarizes our previous work on learning reciprocal relations [@Pahikkala2010]. Let us start with a definition of this type of relation.

A binary relation ${Q}: {\mathcal{V}}^2 \rightarrow [0,1]$ is called a reciprocal relation if for all $({v},{v}') \in {\mathcal{V}}^2$ it holds that ${Q}({v},{v}') = 