Study on the phase transition of the fractal scale-free networks

Based on the Ising spin, the phase transition on fractal scale-free networks with tree-like skeletons is studied, where the loops are generated by local links. The degree distribution of the tree-like skeleton satisfies the power-law form P(k)~k^-δ. It is found that when δ ≥ 3, the renormalized scale-free network will have the same degree distribution as the original network. For a special case of δ=4.5, a ferromagnetic to paramagnetic transition is found and the critical temperature is determined by the box-covering renormalization method. By keeping the structure of the fractal scale-free network constant, the numerical relationship between the critical temperature and the network size is found, which is the form of power law.