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 $ψ$ of the root clusters to show that $ψ$ 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 $n_s \propto s^{-τ}$, where $τ$ satisfies the scaling relation $τ=1+ψ^{-1}$. In addition the critical exponent $β(\tilde{p})$ of the order parameter varies as $\tilde{p}$, from $β\simeq 0.164694$ at $\tilde{p}=0$ to infinity at $\tilde{p}=\tilde{p}_c=5/32$.

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