2008/05/31 by Stefan Boettcher, S. Boettcher, B. Goncalves +3 · 1 citation
Mathematics · Physics and Astronomy · #Complex Network Analysis Techniques #Graph theory and applications #Theoretical and Computational Physics #cond-mat.dis-nn
paper · pdf · doi:10.1088/1751-8113/41/33/335003
published as Journal of Physics A: Math. Theo. 41, 335003 (2008) · 15 pages, revtex; published version; find related material at http://www.physics.emory.edu/faculty/boettcher/
openalex publication_date 2008/07/17 · arxiv created 2008/07/23 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30
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 with the system size, and random walks on HN3 exhibit super-diffusion with an anomalous exponent d w = 2 − log 2 (ϕ) ≈ 1.306, where 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 μ = 2 − 1/ d w . It is shown explicitly that recurrence properties on HN3 depend on the starting site.