Generating-function approach for bond percolations in hierarchical networks

Original · EN

We study bond percolations on hierarchical scale-free networks with the open bond probability of the shortcuts p and that of the ordinary bonds p. The system has a critical phase in which the percolating probability P takes an intermediate value 0<P<1. Using generating function approach, we calculate the fractal exponent ψ of the root clusters to show that ψ varies continuously with p in the critical phase. We confirm numerically that the distribution nₛ of cluster size s in the critical phase obeys a power law nₛ ∝ s⁻τ, where τ satisfies the scaling relation τ=1+ψ⁻¹. In addition the critical exponent β(p) of the order parameter varies as p, from β≃ 0.164694 at p=0 to infinity at p=pc=5/32.