Generating-function approach for bond percolation in hierarchical networks

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 n_s of cluster size s in the critical phase obeys a power law n_s 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= p_c =5/32.