  We investigate the spatiotemporal dynamics of a network of coupled nonlinear
oscillators, modeled by sine circle maps, with varying degrees of randomness in
coupling connections. We show that the change in the basin of attraction of the
spatiotemporal fixed point due to varying fraction of random links $p$, is
crucially related to the nature of the local dynamics. Even the qualitative
dependence of spatiotemporal regularity on $p$ changes drastically as the
angular frequency of the oscillators change, ranging from monotonic increase or
monotonic decrease, to non-monotonic variation. Thus it is evident here that
the influence of random coupling connections on spatiotemporal order is highly
non-universal, and depends very strongly on the nodal dynamics.
