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 Cayley 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, and provide a number of examples.