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