Patchy percolation on a hierarchical network with small-world bonds

The bond-percolation properties of the recently introduced Hanoi networks are analyzed with the renormalization group. Unlike scale-free networks, these networks 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 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 nonuniversal behavior.