Comment on “Ising model on the scale-free network with a Cayley-tree-like structure”

A clarification is due about the paper by Hasegawa and Nemoto [Phys. Rev. E 75, 026105 (2007)], where a clear distinction between the Zeta and Zipf power-law distributions offers an alternative interpretation of the behavior of susceptibility of the model at hand. More precisely, their conclusion that susceptibility diverges for this scale-free network model with power-law distribution $P\left(k\right)\sim k^{-\gamma }$ for the coordination number $k$ for all temperatures, for values of exponent $\gamma \le 4$ (as observed in real networks), stems from the (infinite domain) Zeta distribution power-law assumption for the coordination number distribution. On the other hand, by assuming the Zipf power-law distribution (with an arbitrary finite upper bound on the coordination number), the susceptibility is well behaved, diverges in the interval $0\le T<T_S$, and is finite for $T\ge T_S$, where $T_S$ depends on $P\left(k\right)$.

Figure 1

Scaled zero-field susceptibility for $\gamma =2.7$, calculated using formula (1), for several network sizes $n=64,256,1024$. The full line represents the limiting curve $\text{ln}\left(\alpha t^2\right)$.