  Using the theory of large deviations, macroscopic fluctuation theory provides
a framework to understand the behaviour of non-equilibrium dynamics and steady
states in diffusive systems. We extend this framework to a minimal model of
non-equilibrium non-diffusive system, specifically an open linear network on a
finite graph. We explicitly calculate the dissipative bulk and boundary forces
that drive the system towards the steady state, and non-dissipative bulk and
boundary forces that drives the system in orbits around the steady state. Using
the fact that these forces are orthogonal in a certain sense, we provide a
decomposition of the large-deviation cost into dissipative and non-dissipative
terms. We establish that the purely non-dissipative force turns the dynamics
into a Hamiltonian system. These theoretical findings are illustrated by
numerical examples.
