Critical and Nonpercolating Phases in Bond Percolation on the Song-Havlin-Makse Network

We investigate bond percolation on the Song-Havlin-Makse (SHM) network, a scale-free tree with a tunable degree exponent and dimensionality. Using a generating function approach, we analytically derive the average size and the fractal exponent of the root cluster for deterministic cases. Our analysis reveals that bond percolation on the SHM network remains in a nonpercolating phase for all $p < 1$ when the network is fractal (i.e., finite-dimensional), whereas it exhibits a critical phase, where the cluster size distribution follows a power-law with a $p$-dependent exponent, throughout the entire range of $p$ when the network is small-world (i.e., infinite-dimensional), regardless of the specific dimensionality or degree exponent. The analytical results are in excellent agreement with Monte Carlo simulations.