looks very similar to the one derived for reciprocal relations. Let us again consider the Kronecker product for $\Psi$, then one obtains for $\Phi_S$ the kernel $K_{\otimes S}^{\Phi}$ given by $K_{\otimes S}^{\Phi}({e},{\overline{{e}}})=$ $$\begin{aligned}
2 \big(K^{\phi}({v},{\overline{{v}}}) K^{\phi}({v}',{\overline{{v}}}') + K^{\phi}({v},{\overline{{v}}}') K^{\phi}({v}',{\overline{{v}}})\big) \,.\end{aligned}$$ Thus, the substraction of kernels in the reciprocal case becomes an addition of kernels in the symmetric case. The above kernel has been used for predicting protein-protein interactions in bioinformatics [@Ben-Hur2005] and it has been theoretically analyzed in [@Hue2010]. More specifically, for some methods one has shown in the latter paper that enforcing symmetry in the kernel function yields identical results as adding every edge twice to the dataset, by taking each of the two nodes once as first element of the edge. Unlike many existing kernel-based methods for pairwise data, the models obtained with these kernels are able to represent any reciprocal or symmetric relation respectively, without imposing additional transitivity properties of the relations.

We also remark that for symmetry as well, one can prove that the Kronecker product edge kernel yields a model that is flexible enough to represent any type of underlying relation.

Let $$S({\mathcal{V}}^2)=\{t \mid t\in C({\mathcal{V}}^2),t({v},{v}')=t({v}',{v})\}$$ be the space of all continuous symmetric relations from 