A ug 2 00 5 Mixed algorithms in the Ising model on directed Barabási - Albert networks

On directed Barabási-Albert networks with two and seven neighbours selected by each added site, the Ising model does not seem to show a spontaneous magnetisation. Instead, the decay time for flipping of the magnetisation follows an Arrhenius law for Metropolis and Glauber algorithms, but for Wolff cluster flipping the magnetisation decays exponentially with time. On these networks the magnetisation behaviour of the Ising model, with Glauber, HeatBath, Metropolis, Wolf or Swendsen-Wang algorithm competing against Kawasaki dynamics , is studied by Monte Carlo simulations. We show that the model exhibits the phenomenon of self-organisation (= stationary equilibrium) defined in [8] when Kawasaki dynamics is not dominant in its competition with Glauber, HeatBath and Swendsen-Wang algorithms. Only for Wolff cluster flipping the magnetisa-tion, this phenomenon occurs after an exponentially decay of magnetisation with time. The Metropolis results are independent of competition. We also study the same process of competition described above but with Kawasaki dynamics at the same temperature as the other algorithms. The obtained results are similar for Wolff cluster flipping, Metropolis and Swendsen-Wang algorithms but different for HeatBath. Introduction Sumour and Shabat [1, 2] investigated Ising models on directed Barabási-Albert networks [3] with the usual Glauber dynamics. No spontaneous mag-netisation was found, in contrast to the case of undirected Barabási-Albert networks [4, 5, 6] where a spontaneous magnetisation was found below a critical temperature which increases logarithmically with system size. More recently, Lima and Stauffer [7] simulated directed square, cubic and hyper-cubic lattices in two to five dimensions with heat bath dynamics in order to separate the network effects form the effects of directedness. They also compared different spin flip algorithms, including cluster flips [9], for Ising-Barabási-Albert networks. They found a freezing-in of the magnetisation