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The Ising model with ferromagnetic couplings on the Hanoi networks isanalyzed with an exact renormalization group. In particular, the fixed-pointsare determined and the renormalization-group flow for certain initialconditions is analyzed. Hanoi networks combine a one-dimensional latticestructure with a hierarchy of small-world bonds to create a mix of geometricand mean-field properties. Generically, the small-world bonds result innon-universal behavior, i.e. fixed points and scaling exponents that depend ontemperature and the initial choice of coupling strengths. It is shown that adiversity of different behaviors can be observed with seemingly small changesin the structure of the networks. Defining interpolating families of suchnetworks, we find tunable transitions between regimes with power-law andcertain essential singularities in the critical scaling of the correlationlength, similar to the so-called inverted Berezinskii-Kosterlitz-Thoulesstransition previously observed only in scale-free or dense networks.

Fixed Point Properties of the Ising Ferromagnet on the Hanoi Networks