We examine Ising models with heat-bath dynamics on directed networks. Oursimulations show that Ising models on directed triangular and simple cubiclattices undergo a phase transition that most likely belongs to the Isinguniversality class. On the directed square lattice the model remainsparamagnetic at any positive temperature as already reported in some previousstudies. We also examine random directed graphs and show that contrary toundirected ones, percolation of directed bonds does not guarantee ferromagneticordering. Only above a certain threshold a random directed graph can supportfinite-temperature ferromagnetic ordering. Such behaviour is found also forout-homogeneous random graphs, but in this case the analysis of magnetic andpercolative properties can be done exactly. Directed random graphs also differfrom undirected ones with respect to zero-temperature freezing. Only at lowconnectivity they remain trapped in a disordered configuration. Above a certainthreshold, however, the zero-temperature dynamics quickly drives the modeltoward a broken symmetry (magnetized) state. Only above this threshold, whichis almost twice as large as the percolation threshold, we expect the Isingmodel to have a positive critical temperature. With a very good accuracy, thebehaviour on directed random graphs is reproduced within a certain approximatescheme.
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