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 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_2ϕ= 1.306..., where ϕ= (\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 μ= 2 -1/d_w . It is shown explicitly that recurrence properties on HN3 depend on the starting site.

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

References in corpus (5)

• First-passage times in complex scale-invariant media
• Deterministic Scale-Free Networks
• Inverted Berezinskii-Kosterlitz-Thouless Singularity and High-Temperature Algebraic Order in an Ising Model on a Scale-Free Hierarchical-Lattice Small-World Network
• Maximal planar scale-free Sierpinski networks with small-world effect and power-law strength-degree correlation
• Hierarchical, Regular Small-World Networks

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• Chaotic Spin Correlations in Frustrated Ising Hierarchical Lattices
• Critical Percolation Phase and Thermal BKT Transition in a Scale-Free Network with Short-Range and Long-Range Random Bonds
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