In the mathematical study of reaction networks, the most classical results are those pertaining to models that have a deficiency of zero, which, loosely, is a linear independence property of the associated reaction vectors. In particular, the classical results for deterministic models, which date back to the early 1970s, focus on models with a deficiency of zero andconnect them to detailed-balanced models. Moreover, the largest open conjecture related to deterministic models, the so-called Global Attractor Conjecture, focuses on models with a deficiency of zero. In the stochastic reaction network setting it has been shown that a deficiency of zero together with a certain connectivity property of the associated reaction network implies that the stationary distribution of the model is a product of Poissons. Given that deficiency zero models play such a significant role in the mathematical study of reaction networks, a natural question is : how prevalent are they? In order to answer this question, we consider reaction networks under an Erdos-Renyi random graph framework. In particular, we start with n species, and then let our possible vertices be all zeroth, first, and second order complexes that can be produced from the n species. Edges, or reversible reactions, between two arbitrary vertices then occur independently with probability p_n. We establish a function, r(n), termed athreshold function , such that the probability of the random network being deficiency zero converges to 1 if p_n \ll r(n ){r(n)}}\to \to  and converges to 0 if p_n \gg{r(n)}}\to \to  r(n)  .
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 .
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In the mathematical study of reaction networks, the most classical results are those pertaining to models that have a deficiency of zero, which, loosely, is a linear independence property of the associated reaction vectors. In particular, the classical results for deterministic models, which date back to the early 1970s, focus on models with a deficiency of zero andconnect them to detailed-balanced models. Moreover, the largest open conjecture related to deterministic models,
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.,
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so-called Global Attractor Conjecture, focuses on models with a deficiency of zero. In the stochastic reaction network setting it has been shown that
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
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together with a certain connectivity property of the associated reaction network implies that the stationary distribution of the model is a product of Poissons. Given that deficiency zero models play such a significant role in the mathematical study of reaction networks, a natural question is : how prevalent are they? In order to answer this question, we consider reaction networks under an Erdos-Renyi random graph framework. In particular, we start with n species, and then let our possible vertices be all zeroth, first, and second order complexes that can be produced from the n species. Edges, or reversible reactions, between two arbitrary vertices then occur independently with probability p_n. We establish a function, r(n), termed a
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threshold function
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, such
, 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
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the random
a random binary
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p_n \ll r(n )
\frac{p_n
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0, as n
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p_n \gg
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A
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