1 - {Q}({v}',{v})$.

A binary relation ${h}: {\mathcal{V}}^2 \rightarrow {\mathbb{R}}$ is called an antisymmetric relation if for all $({v},{v}') \in {\mathcal{V}}^2$ it holds that ${h}({v},{v}') = - {h}({v}',{v})$.

For reciprocal and antisymmetric relations, every edge ${e}=({v},{v}')$ in a multi-graph like Figure \[fig:examples\] induces an unobserved invisible edge ${e}_R = ({v}',{v})$ with appropriate weight in the opposite direction. The transformation operator $\nabla$ transforms an antisymmetric relation into a reciprocal relation. Applications of reciprocal relations arise here in domains such as preference learning, game theory and bioinformatics for representing preference relations, choice probabilities, winning probabilities, gene regulation, etc. The weight on the edge defines the real direction of such an edge. If the weight on the edge ${e}= ({v},{v}')$ is higher than 0.5, then the direction is from $v$ to $v'$, but when the weight is lower than 0.5, then the direction should be interpreted as inverted, for example, the edges from $A$ to $C$ in Figures \[fig:examples\] (a) and (e) should be interpreted as edges starting from $A$ instead of $C$. If the relation is $3$-valued as $Q: {\mathcal{V}}^2 \rightarrow \{0,1/2,1\}$, then we end up with a three-class ordinal regression setting instead of an ordinary regression setting.

Interestingly, reciprocity can be 