Before revision:  Specifically, given a randomly generated binary reaction network with n species, with an edge between two arbitrary vertices occurring independently with probability p_n, we established the threshold function r(n)=1{n^3} such that the probability of the random network being deficiency zero converges to 1 if p_n \ll r(n){r(n)}}\to  and converges to 0 if p_n \gg r(n){r(n)}}\to , as n \to \infty.
After revision:  Specifically, given a randomly generated binary reaction network with n species, with an edge between two arbitrary vertices occurring independently with probability p_n, we established the threshold function r(n)=1{n^3} such that the probability of the random network being deficiency zero converges to 1 if p_n \ll r(n){r(n)}}\to 0 and converges to 0 if p_n \gg r(n){r(n)}}\to , as n \to \infty.