In the mathematical study of reaction networks, many of the classical results pertain to models that have a deficiency of zero  . In particular,  for deterministic models it is well known that weak reversibility and a deficiency zero of the reaction network imply that the model is complex balanced . In the stochastic setting it is known that weak reversibility and a deficiency of zero imply the existence of a stationary distribution that 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 complexes then occur independently with probability p_n. We establish a function, r(n), termed a threshold function , such that the probability of the random network being deficiency zero converges to 1 if p_n \ll r(n) and converges to 0 if p_n \gg r(n).
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 and connect 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 a threshold function , such that the probability of the random network being deficiency zero converges to 1 if p_n \ll r(n) and converges to 0 if p_n \gg r(n).
R
many of the classical results pertain
the most classical results are those pertaining
A
, which, loosely, is a linear independence property of the associated reaction vectors
A
the classical results
R
it is well known that weak reversibility and a deficiency zero of the reaction network imply that the model is complex balanced
, which date back to the early 1970s, focus on models with a deficiency of zero and connect 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
R
setting it is known that weak reversibility and
reaction network setting it has been shown that
R
imply the existence of a stationary distribution that
together with a certain connectivity property of the associated reaction network implies that the stationary distribution of the model
A
:
R
complexes
vertices
R
threshold function
threshold function