Phase transitions for a rock-scissors-paper model with long-range-directed interactions

We have investigated a rock-scissors-paper model with long-range-directed interactions in two dimensions where every site has four outgoing links but a fraction q of the outgoing links to the nearest neighbour sites are rewired to other long-distance sites chosen randomly and the lattice structure is replaced again after a Monte Carlo step. It is found that, with q increasing, the system changes from a three species coexistence self-organizing state to a global oscillation state and then to one of the homogeneous states. However when q exceeds a third threshold value, the system returns to a self-organizing state. When we restrict the maximum number of ingoing links of a site to four, the last self-organizing state disappears, the system stays in the homogeneous state forever after q exceeds the second threshold value. And when we restrict the maximum number of ingoing links of a site to five or six, the system exhibits a transition from the homogeneous state to a global oscillation state again and then to the last self-organizing state with q increasing. The comparison of results on different networks suggests that the sites with zero ingoing links should play a significant role in the emergences of the later self-organizing state and the subsequent global oscillation