Absence of the non-percolating phase for percolation on the non-planar Hanoi network

Abstract

We investigate bond percolation on the non-planar Hanoi network (HN-NP), which was studied in [Boettcher et al. Phys. Rev. E 80 (2009) 041115]. We calculate the fractal exponent of a subgraph of the HN-NP, which gives a lower bound for the fractal exponent of the original graph. This lower bound leads to the conclusion that the original system does not have a non-percolating phase, where only finite size clusters exist, for p>0, or equivalently, that the system exhibits either the critical phase, where infinitely many infinite clusters exist, or the percolating phase, where a unique giant component exists. Monte Carlo simulations support our conjecture.

arXiv comment5 pages, 4 figures

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• A general model of hierarchical fractal scale-free networks
• Hierarchical scale-free network is fragile against random failure
• Classification of critical phenomena in hierarchical small-world networks
• Profile and scaling of the fractal exponent of percolations in complex networks
• From explosive to infinite-order transitions on a hyperbolic network
• Real-Space Renormalization Group for Spectral Properties of Hierarchical Networks