  We examine Ising models with heat-bath dynamics on directed networks. Our
simulations show that Ising models on directed triangular and simple cubic
lattices undergo a phase transition that most likely belongs to the Ising
universality class. On the directed square lattice the model remains
paramagnetic at any positive temperature as already reported in some previous
studies. We also examine random directed graphs and show that contrary to
undirected ones, percolation of directed bonds does not guarantee ferromagnetic
ordering. Only above a certain threshold a random directed graph can support
finite-temperature ferromagnetic ordering. Such behaviour is found also for
out-homogeneous random graphs, but in this case the analysis of magnetic and
percolative properties can be done exactly. Directed random graphs also differ
from undirected ones with respect to zero-temperature freezing. Only at low
connectivity they remain trapped in a disordered configuration. Above a certain
threshold, however, the zero-temperature dynamics quickly drives the model
toward a broken symmetry (magnetized) state. Only above this threshold, which
is almost twice as large as the percolation threshold, we expect the Ising
model to have a positive critical temperature. With a very good accuracy, the
behaviour on directed random graphs is reproduced within a certain approximate
scheme.
