Abstract: In statistical thermodynamics, a Markov chain models an equilibrium system if all probability fluxes across pairs of forward and backward edges vanish at the steady state. Equilibrium systems are energetically closed. Most life processes are not energetically closed. The continual use of energy to perform a function is the defining hallmark of active matter. As such, there is great interest in non-equilibrium systems where energy is exchanges maintain nonvanishing, cyclic probability currents. The Helmholtz-Hodge decomposition (HHD) can be used to separate the work to cross each edge in a discrete-state Markov chain into a conservative component associated with internal energy, and a rotational component associated with external energy sources. We consider the near equilibrium case, when driving rotation is weak, and introduce a formal expansion of the steady state distribution and currents as a Taylor series. Each term satisfies a recursively defined HHD. We derive a nonzero lower bound on the radius of convergence of the expansion. We use the expansion to prove classical results from linear thermodynamics such as reciprocity. We also use the expansion to explore situations in which the steady state is independent of driving rotation at all orders, or up to a given order, to show that some reciprocity extends past first order, and to relate the thermal efficiency of the system and the amount the steady state is perturbed. Our approach applies to any Markov chain with microscopic reversibility.