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 edgelabelled directed graph encoding the jump vectors of an associated continuous time Markov chain on an ambient space N d0 . An open question is whether the ambient space 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 N d 0 that generates continuous time Markov chains taking values in N d 0 , 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.