Percolation thresholds on finitely ramified fractals

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.

Percolation in networks with voids and bottlenecks

A general method is proposed for predicting the asymptotic percolation threshold of networks with bottlenecks, in the limit that the subnet mesh size goes to zero. The validity of this method is tested for bond percolation on filled checkerboard and "stack-of-triangle" lattices. Thresholds for the checkerboard lattices of different mesh sizes are estimated using the gradient percolation method, while for the triangular system they are found exactly using the triangle-triangle transformation. The values of the thresholds approach the asymptotic values of 0.64222 and 0.53993, respectively, as the mesh is made finer, consistent with a direct determination based upon the predicted critical corner-connection probability.