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.