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We consider the ordering dynamics of the Ising model on a square latticewhere an additional fixed number of bonds connect any two sites chosenrandomly. The total number of shortcuts added is controlled by two parameters$p$ and $\alpha$. The structural properties of the network are investigatedwhich show that that the small world behaviour is obtained along the line$\alpha=\frac{\ln (N/2p)}{\ln N}$, which separates regions with ultra smallworld like behaviour and short ranged lattice like behaviour. We obtain a richphase diagram in the $p-\alpha$ plane showing the existence of different typesof active and absorbing states to which Ising model evolves and theirboundaries.

Active-absorbing phase transition and small world behaviour in Ising model on finite addition type networks in two dimensions