  A reaction network consists of a finite number of species, which interact
through predefined reaction channels. Traditionally such networks were modeled
deterministically, but it is now well-established that when reactant copy
numbers are small, the random timing of the reactions create internal noise
that can significantly affect the macroscopic properties of the system. To
understand the role of noise and quantify its effects, stochastic models are
necessary. In the stochastic setting, the population is described by a
probability distribution, which evolves according to a set of ordinary
differential equations known as the Chemical Master Equation (CME). This set is
infinite in most cases making the CME practically unsolvable. In many
applications, it is important to determine if the solution of a CME has a
globally attracting fixed point. This property is called ergodicity and its
presence leads to several important insights about the underlying dynamics. The
goal of this paper is to present a simple procedure to verify ergodicity in
stochastic reaction networks. We provide a set of simple linear-algebraic
conditions which are sufficient for the network to be ergodic. In particular,
our main condition can be cast as a Linear Feasibility Problem (LFP) which is
essentially the problem of determining the existence of a vector satisfying
certain linear constraints. The inherent scalability of LFPs make our approach
efficient, even for very large networks. We illustrate our procedure through an
example from systems biology.
