  The Ising model with ferromagnetic couplings on the Hanoi networks is
analyzed with an exact renormalization group. In particular, the fixed-points
are determined and the renormalization-group flow for certain initial
conditions is analyzed. Hanoi networks combine a one-dimensional lattice
structure with a hierarchy of small-world bonds to create a mix of geometric
and mean-field properties. Generically, the small-world bonds result in
non-universal behavior, i.e. fixed points and scaling exponents that depend on
temperature and the initial choice of coupling strengths. It is shown that a
diversity of different behaviors can be observed with seemingly small changes
in the structure of the networks. Defining interpolating families of such
networks, we find tunable transitions between regimes with power-law and
certain essential singularities in the critical scaling of the correlation
length, similar to the so-called inverted Berezinskii-Kosterlitz-Thouless
transition previously observed only in scale-free or dense networks.
