Geometry and dynamics for hierarchical regular networks
Abstract
The recently introduced hierarchical regular networks HN3 and HN4 are analyzed in detail. We use renormalization group arguments to show that HN3, a 3regular planar graph, has a diameter growing as \sqrt{N} with the system size, and random walks on HN3 exhibit superdiffusion with an anomalous exponent d_{w} = 2  log_{2}(phi) ≈ 1.306, where \phi=(\sqrt{5}+1)/2=1.618\ldots is the 'golden ratio.' In contrast, HN4, a nonplanar 4regular 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 firstreturn probability with a power law tail characterized by an exponent μ = 2  1/d_{w}. It is shown explicitly that recurrence properties on HN3 depend on the starting site.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 August 2008
 DOI:
 10.1088/17518113/41/33/335003
 arXiv:
 arXiv:0805.3013
 Bibcode:
 2008JPhA...41G5003B
 Keywords:

 Condensed Matter  Disordered Systems and Neural Networks
 EPrint:
 15 pages, revtex