  We analyze the phase transitions that emerge from the recursive design of
certain hyperbolic networks that includes, for instance, a discontinuous
("explosive") transition in ordinary percolation. To this end, we solve the
$q$-state Potts model in the analytic continuation for non-integer $q$ with the
real-space renormalization group. We find exact expressions for this
one-parameter family of models that describe the dramatic transformation of the
transition. In particular, this variation in $q$ shows that the discontinuous
transition is generic in the regime $q<2$ that includes percolation. A
continuous ferromagnetic transition is recovered in a singular manner only for
the Ising model, $q=2$. For $q>2$ the transition immediately transforms into an
infinitely smooth order parameter of the Berezinskii-Kosterlitz-Thouless (BKT)
type.
