  We study a class of Gibbs measures of classical particle spin systems with
spin space $S=\mathbb{R}^{m}$ and unbounded pair interaction, living on a
metric graph given by a typical realization $\gamma $ of a random point process
in $\mathbb{R}^{n}$. Under certain conditions of growth of pair- and
self-interaction potentials, we prove that the set $\mathcal{G}(S^{\gamma})$ of
all such Gibbs measures is not empty for almost all $\gamma $, and study
support properties of $\nu_{\gamma}\in \mathcal{G}(S^{\gamma})$. Moreover we
show the existence of measurable maps (selections) $\gamma \mapsto
\nu_{\gamma}$ and derive the corresponding averaged moment estimates.
