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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 Erdős-Rényi 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 ≪ r(n) and converges to 0 if p n ≫ r(n).

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