  We propose a general classification of nonequilibrium steady states in terms
of their stationary probability distribution and the associated probability
currents. The stationary probabilities can be represented graph-theoretically
as directed labelled trees; closing a single loop in such a graph leads to a
representation of probability currents. This classification allows us to
identify all choices of transition rates, based on a master equation, which
generate the same nonequilibrium steady state. We explore the implications of
this freedom, e.g., for entropy production.
