Geometry and dynamics for hierarchical regular networks

The recently introduced hierarchical regular networks HN3 and HN4 are analyzed in detail. We use renormalization group arguments to show that HN3, a 3-regular planar graph, has a diameter growing as root N with the system size, and random walks on HN3 exhibit super-diffusion with an anomalous exponent d(w) = 2 - log(2)(phi) approximate to 1.306, where phi = (root 5+ 1)/2 = 1.618 ... is the 'golden ratio.' In contrast, HN4, a non-planar 4-regular graph, has a diameter that grows slower than any power of N, yet, faster than any power of ln N. In an annealed approximation we can show that diffusive transport on HN4 occurs ballistically (d(w) = 1). Walkers on both graphs possess a first-return probability with a power law tail characterized by an exponent mu = 2 - 1/d(w). It is shown explicitly that recurrence properties on HN3 depend on the starting site.