This paper is motivated by demands in stochastic reaction network theory. The $Q$-matrix of a stochastic reaction network can be derived from the reaction graph, an edge-labelled directed graph encoding the jump vectors of an associated continuous time Markov chain on an ambient space $mathbb{N}^d_0$. An open question is whether the ambient space $mathbb{N}^d_0$ can be decomposed into neutral, trapping, and escaping states, and open and closed communicating classes, based on the reaction graph alone. We characterize the structure of the state space of a $Q$-matrix on $mathbb{N}^d_0$ that generates continuous time Markov chains taking values in $mathbb{N}_0^d$, in terms of the set of jump vectors and their corresponding transition rate functions. We also define structural equivalence of two $Q$-matrices, and provide sufficient conditions for structural equivalence. Such stochastic processes are abundant in applications. We apply our results to stochastic reaction networks, a Lotka-Volterra model in ecology, the EnvZ-OmpR system in systems biology, and a class of extended branching processes, none of which are birth-death processes.
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