Geometry and Dynamics for Hierarchical Regular Networks

Details Geometry and Dynamics for Hierarchical Regular Networks Geometry and Dynamics for Hierarchical Regular Networks

2008-05-20

arxiv.org/abs/0805.3013v2

Journal of Physics A: Math. Theo. 41, 335003 (2008)

Physics

Condensed Matter

Disordered Systems and Neural Networks

15 pages, revtex; published version; find related material at http://www.physics.emory.edu/faculty/boettcher/

Scientific paper

10.1088/1751-8113/41/33/335003

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 \sqrt{N} with the system size, and random walks on HN3 exhibit super-diffusion with an anomalous exponent d_w = 2 - \log_2\phi = 1.306..., where \phi = (\sqrt{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, fast 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.

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