First-order transition in small-world networks

Small-world networks are regular lattices with a fraction $p$ of long-range bonds. For small $p$, minimum-distance properties of random graphs appear for sizes $L>L^*(p)$. It has been recently argued that this is a crossover phenomenon with $L^*\sim p^{-\tau}$, while other authors claimed that it is a second-order transition at $p=0$, with a diverging correlation length $\xi \sim p^{-\nu}$ and $\nu=1/d$. We show that no diverging correlation length can be defined, so that this transition is first-order. We find that $\tau= 1/d$ in one to four dimensions, and justify this by the same rescaling procedure as Newman and Watts.