Deficiency zero for random reaction networks under a stochastic block model framework

Abstract: Deficiency zero is an important network structure and has been the focus of many celebrated results within reaction network theory. In our previous paper \textit{Prevalence of deficiency zero reaction networks in an Erd\H os-R\'enyi framework}, we provided a framework to quantify the prevalence of deficiency zero among randomly generated reaction networks. 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 $r(n)=\frac{1}{n^3}$ such that the probability of the random network being deficiency zero converges to 1 if $\frac{p_n}{r(n)}\to 0$ and converges to 0 if $\frac{p_n}{r(n)}\to\infty$, as $n \to \infty$. With the base Erd\H os-R\'enyi framework as a starting point, the current paper provides a significantly more flexible framework by weighting the edge probabilities via control parameters $\alpha_{i,j}$, with $i,j\in {0,1,2}$ enumerating the types of possible vertices (zeroth, first, or second order). The control parameters can be chosen to generate random reaction networks with a specific underlying structure, such as "closed" networks with very few inflow and outflow reactions, or "open" networks with abundant inflow and outflow. Under this new framework, for each choice of control parameters ${\alpha_{i,j}}$, we establish a threshold function $r(n,{\alpha_{i,j}})$ such that the probability of the random network being deficiency zero converges to 1 if $\frac{p_n}{r(n,{\alpha_{i,j}})}\to 0$ and converges to 0 if $\frac{p_n}{r(n,{\alpha_{i,j}})}\to \infty$.