this type of domain knowledge.

The notion of ranking representability is powerful for reciprocal relations, because the majority of reciprocal relations satisfy this property, but for symmetric relations it has a rather limited applicability. Ranking representability as defined above cannot represent relations that originate from an underlying metric or similarity measure. For such relations, one needs another connection with its roots in Euclidean metric spaces [@Gower1986].

A symmetric relation ${Q}: {\mathcal{V}}^2 \rightarrow [0,1]$ is called Euclidean representable if there exists a ranking function $f: {\mathcal{V}}\rightarrow {\mathbb{R}}$ such that for all pairs $({v},{v}')
\in {\mathcal{V}}^2$ it holds that $$\begin{aligned}
\label{eq:rrsym}
{Q}({v},{v}') = \nabla (({f}({v}) - {f}({v}'))^T({f}({v}) - {f}({v}'))) \,,\end{aligned}$$ with $\vec{a}^T$ the transpose of a vector $\vec{a}$.

Euclidean representability as defined here basically can be seen as Euclidean embedding or Multidimensional Scaling in a $z$-dimensional space [@Zhang2003]. In its most restrictive form, when $z=1$, it implies that the symmetric relation can be constructed from the Euclidean distance in a one-dimensional space. When such a one-dimensional embedding can be realized, one global ranking of the objects can be found, similar to reciprocal relations. Nevertheless, although models of type (\[eq:rrsym\]) with $z=1$ are sometimes used in graph inference [@Vert2005] and semi-supervised learning [@Belkin2006], 