Generating-function approach for bond percolations in hierarchical networks

We study bond percolations on hierarchical scale-free networks with the open bond probability of the shortcuts $\tilde{p}$ 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 $\psi$ of the root clusters to show that $\psi$ varies continuously with $\tilde{p}$ in the critical phase. We confirm numerically that the distribution $n_s$ of cluster size $s$ in the critical phase obeys a power law $n_s \propto s^{-\tau}$, where $\tau$ satisfies the scaling relation $\tau=1+\psi^{-1}$. In addition the critical exponent $\beta(\tilde{p})$ of the order parameter varies as $\tilde{p}$, from $\beta\simeq 0.164694$ at $\tilde{p}=0$ to infinity at $\tilde{p}=\tilde{p}_c=5/32$.