Susceptibility of the Ising model on the scale-free network with a Cayley tree-like structure

We derive the exact expression for the zero-field susceptibility of each spin of the Ising model on the scale-free (SF) network having the degree distribution P(k)∝k−γ with the Cayley tree-like structure. The system shows that: (i) the zero-field susceptibility of a spin in the interior part diverges below the transition temperature of the SF network with the Bethe lattice-like structure Tc for γ>3, while it diverges at any finite temperature for γ≤3, and (ii) the surface part diverges below the divergence temperature of the SF network with the Cayley tree-like structure Ts for γ>3, while it diverges at any finite temperature for γ≤3.