Small-World Bonds and Patchy Percolation on the Hanoi Network

TL;DR

This paper investigates the bond-percolation properties of Hanoi networks, revealing how hierarchical small-world bonds create a unique 'patchy' order that influences phase transitions and connectivity in complex networks.

Contribution

It provides an analytical study of percolation in Hanoi networks, highlighting the impact of hierarchical small-world bonds on phase behavior and connectivity.

Findings

Hierarchical small-world bonds induce a 'patchy' order.

Percolation exhibits finite probability of spanning clusters, unlike standard lattices.

Network phase behavior varies with the prevalence of long-range bonds.

Abstract

The bond-percolation properties of the Hanoi networks are analyzed with the renormalization group. Unlike scale-free networks, they are meant to provide an analytically tractable interpolation between finite dimensional, lattice-based models and their mean-field limits. In percolation, the hierarchical small-world bonds in the Hanoi networks impose a new form of order by uniting otherwise disconnected, local clusters. This "patchy" order results in merely a finite probability to obtain a spanning cluster for certain ranges of the bond probability, unlike the usual 0-1 transition found on ordinary lattices. The various networks studied here exhibit a range of phase behaviors, depending on the prevalence of those long-range bonds. Fixed points in general exhibit non-universal behavior.