  Reaction networks are commonly used within the mathematical biology and
mathematical chemistry communities to model the dynamics of interacting
species. These models differ from the typical graphs found in random graph
theory since their vertices are constructed from elementary building blocks,
i.e., the species. In this paper, we consider these networks in an Erd\H
os-R\'enyi framework and, under suitable assumptions, derive a threshold
function for the network to have a deficiency of zero, which is a property of
great interest in the reaction network community. Specifically, if the number
of species is denoted by $n$ and if the edge probability is denote by $p_n$,
then we prove that the probability of a random binary network being deficiency
zero converges to 1 if $\frac{p_n}{r(n)}\to 0$, as $n \to \infty$, and
converges to 0 if $\frac{p_n}{r(n)}\to \infty$, as $n \to \infty$, where
$r(n)=\frac{1}{n^3}$.
