Before revision:  We establish a function, r(n), termed athreshold function , such that the probability of the random network being deficiency zero converges to 1 if p_n \ll r(n ){r(n)}}\to \to  and converges to 0 if p_n \gg{r(n)}}\to \to  r(n)  .
After revision:  We establish a function, r(n), termed athreshold function , such that the probability of the random network being deficiency zero converges to 1 if p_n \ll r(n ){r(n)}}\to \to \infty, and converges to 0 if p_n \gg{r(n)}}\to \to r(n) .