Critical Phase of Bond Percolations on Growing Networks

The critical phase of bond percolation on the random growing tree is examined. It is shown that the root cluster grows with the system size $N$ as $N^\psi$ and the mean number of clusters with size $s$ per node follows a power function $n_s \propto s^{-\tau}$ in the whole range of open bond probability $p$. The exponent $\tau$ and the fractal exponent $\psi$ are also derived as a function of $p$ and the degree exponent $\gamma$, and are found to satisfy the scaling relation $\tau=1+\psi^{-1}$. Numerical results with several network sizes are quite well fitted by a finite size scaling for a wide range of $p$ and $\gamma$, which gives a clear evidence for the existence of a critical phase.