the nodes in such graphs can be obtained with topological sorting algorithms. The ranking representable reciprocal relations of Figures \[fig:examples\] (a) and (e) for example yield the global ranking $A \succ B \succ C$. Interestingly, ranking representability of reciprocal relations and symmetric relations can be easily achieved in our framework by simplifying the joint feature mapping $\Psi$. Let $\Psi({v},{v}') = \phi({v})$ such that $K^{\Phi}$ simplifies to $$\begin{aligned}
\arraycolsep=2pt
K_{{f}R}^{\Phi}({e},{\overline{{e}}})&=&K^{\phi}({v},{\overline{{v}}}) + K^{\phi}({v}',{\overline{{v}}}')  -K^{\phi}({v},{\overline{{v}}}') -
K^{\phi}({v}',{\overline{{v}}}) \,, \\
K_{{f}S}^{\Phi}({e},{\overline{{e}}})&=&K^{\phi}({v},{\overline{{v}}}) + K^{\phi}({v}',{\overline{{v}}}') +K^{\phi}({v},{\overline{{v}}}') +
K^{\phi}({v}',{\overline{{v}}}) \,,\end{aligned}$$ when $\Phi({v},{v}') = \Phi_R({v},{v}')$ or $\Phi({v},{v}') = \Phi_S({v},{v}')$, respectively, then the following proposition holds.

The relation $Q: {\mathcal{V}}^2 \rightarrow [0,1]$ given by (\[eq:monmap\]) and $h$ defined by (\[eq:primalmodel\]) with $K^{\Phi} = K_{{f}R}^{\Phi}$ (respectively $K^{\Phi} = K_{{f}S}^{\Phi}$) is a ranking representable reciprocal (respectively symmetric) relation.

The proof directly follows from the fact that for this specific kernel, ${h}({v},{v}')$ can be respectively written as ${f}({v}) - {f}({v}')$ and ${f}({v}) + {f}({v}')$. The kernel $K_{{f}R}^{\Phi}$ has been initially introduced in [@Herbrich2000] for ordinal regression and during the last decade it has been extensively used as a main building block in many kernel-based ranking algorithms. Since ranking representability of reciprocal relations implies strong stochastic transitivity of reciprocal relations, $K_{{f}R}^{\Phi}$ can represent 