Prevalence of deficiency zero reaction networks in an Erdos-Renyi framework

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 $r(n)=\frac{1}{n^3}$.