Here's a contour plot of predicted giant component size vs. σ on the x-axis and δ on the y-axis.
Unfortunately things get a little below 1/4 on the far right, but I'll blame it on numerical inaccuracy. The interesting thing is apparently there's an abrupt (meta-?)phase shift between the smooth σ-varying phase transition for δ < 0.45 or so and the abrupt transition for δ > 0.45. The "NNN.out" files indicate data showing the transition from the giant component comprising less or more than 0.NNN fraction of the vertices in the graph.