  We have investigated the anomalous scaling behaviour of the Ising model on
small-world networks based on 2- and 3-dimensional lattices using Monte Carlo
simulations. Our main result is that even at low $p$, the shift in the critical
temperature $\Delta T_c$ scales as $p^{s}$, with $s \approx 0.50$ for 2-D
systems, $s \approx 0.698$ for 3-D and $s \approx 0.75$ for 4-D. We have also
verified that a $z$-preserving rewiring algorithm still exhibits small-world
effects and yet is more directly comparable with the conventional Ising model;
the small-world effect is due to enhanced long-range correlations and not the
change in effective dimension. We find the critical exponents $\beta$ and $\nu$
exhibit a monotonic change between an Ising-like transition and mean-field
behaviour in 2- and 3-dimensional systems.
