Before revision:  Under this new framework, for each choice of control parameters \{i,j\}, we establish a threshold function r(n,\{i,j\}) such that the probability of the random network being deficiency zero converges to 1 if p_n \ll r(n,\{\alpha_{i,j\}) } {r(n,\{\alpha_{i,j}\})}}\to  and converges to 0 if p_n \gg r(n,\{\alpha_{i,j\})} {r(n,\{\alpha_{i,j}\})}}\to  .
After revision:  Under this new framework, for each choice of control parameters \{i,j\}, we establish a threshold function r(n,\{i,j\}) such that the probability of the random network being deficiency zero converges to 1 if p_n \ll r(n,\{\alpha_{i,j\}) } \frac{p_n{r(n,\{\alpha_{i,j}\})}}\to and converges to 0 if p_n \gg r(n,\{\alpha_{i,j\})} {r(n,\{\alpha_{i,j}\})}}\to .