Geometry and Dynamics for Hierarchical Regular Networks

TL;DR

This paper analyzes hierarchical regular networks HN3 and HN4, revealing their geometric properties and anomalous diffusion behaviors using renormalization group methods, with implications for understanding complex network dynamics.

Contribution

It provides a detailed analysis of HN3 and HN4 networks, showing their diameters and diffusion properties, and introduces new insights into their recurrence and transport behaviors.

Findings

HN3 has diameter growing as rac{1}{2} with system size

Super-diffusive behavior with anomalous exponent d_w=1.306 on HN3

Ballistic diffusion on HN4 with d_w=1

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_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.