  We study Gibbs distributions of spins taking values in a general compact
Polish space, interacting via a pair potential along the edges of a generalized
random graph with a given asymptotic weight distribution $P$, obtained by
annealing over the random graph distribution.
  First we prove a variational formula for the corresponding annealed pressure
and provide criteria for absence of phase transitions in the general case.
  We furthermore study classes of models with second order phase transitions
which include rotation-invariant models on spheres and models on intervals, and
classify their critical exponents. We find critical exponents which are
modified relative to the corresponding mean-field values when $P$ becomes too
heavy-tailed, in which case they move continuously with the tail-exponent of
$P$. For large classes of models they are the same as for the Ising model
treated in \cite{DomGiaGibHofPri16}. On the other hand, we provide conditions
under which the model is in a different universality class, and construct an
explicit example of such a model on the interval.
