Hierarchical networks of regular degree

Two new networks are introduced that are recursively constructed but retain a fixed, regular degree. They consist of a one-dimensional lattice backbone overlayed by a hierarchical sequence of long-distance links. Both networks, one 3-regular and the other 4-regular, lead to distinct behaviors, as revealed by renormalization group studies. The 3-regular networks are planar, have a diameter , and lead to super-diffusion with an exact, anomalous exponent dw=1.3057581…, but possesses only a trivial fixed point Tc=0 for the Ising ferromagnet. In turn, the 4-regular networks are non-planar, have a diameter of , exhibit “ballistic” diffusion (dw=1), and a Tc>0.