  We study the time-averaged flow in a model of particles that randomly hop on
a finite directed graph. In the limit as the number of particles and the time
window go to infinity but the graph remains finite, the large-deviation rate
functional of the average flow is given by a variational formulation involving
paths of the density and flow. We give sufficient conditions under which the
large deviations of a given time averaged flow is determined by paths that are
constant in time. We then consider a class of models on a discrete ring for
which it is possible to show that a better strategy is obtained producing a
time-dependent path. This phenomenon, called a dynamical phase transition, is
known to occur for some particle systems in the hydrodynamic scaling limit,
which is thus extended to the setting of a finite graph.
