Gradowski T, Krawiecki A
Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL-00-662 Warsaw, Poland.
Phys Rev E. 2020 Aug;102(2-1):022314. doi: 10.1103/PhysRevE.102.022314.
The q-voter model with independence is investigated on multiplex networks with full overlap of nodes in the layers. The layers are various complex networks corresponding to different levels of social influence. Detailed studies are performed for the model on multiplex networks with two layers with identical degree distributions, obeying the LOCAL&AND and GLOBAL&AND spin update rules differing by the way in which the q-lobbies of neighbors within different layers exert their joint influence on the opinion of a given agent. Homogeneous pair approximation is derived for a general case of a two-state spin model on a multiplex network and its predictions are compared with results of mean-field approximation and Monte Carlo simulations of the above-mentioned q-voter model with independence for a broad range of parameters. As the parameter controlling the level of agents' independence is changed ferromagnetic phase transition occurs which can be first- or second-order, depending on the size of the lobby q. Details of this transition, e.g., position of the critical points, critical exponents and the width of the possible hysteresis loop, depend on the topology and other features of the layers, in particular on the mean degree of nodes in the layers which is directly predicted by the homogeneous pair approximation. If the mean degree of nodes is substantially larger than the size of the q-lobby good agreement is obtained between numerical results and theoretical predictions based on the homogeneous pair approximation concerning the order and details of the ferromagnetic transition. In the case of the model on multiplex networks with layers in the form of homogeneous Erdős-Rényi and random regular graphs as well as weakly heterogeneous scale-free networks this agreement is quantitative, while in the case of layers in the form of strongly heterogeneous scale-free networks it is only qualitative. If the mean degree of nodes is small and comparable with q predictions of the homogeneous pair approximation are in general even qualitatively wrong.
在各层节点完全重叠的多重网络上研究具有独立性的q-投票者模型。各层是对应于不同社会影响水平的各种复杂网络。针对具有相同度分布的两层多重网络上的该模型进行了详细研究，该模型遵循LOCAL&AND和GLOBAL&AND自旋更新规则，这两种规则的不同之处在于不同层内邻居的q-游说团体对给定主体意见施加联合影响的方式。推导了多重网络上两态自旋模型一般情况的均匀对近似，并将其预测结果与上述具有独立性的q-投票者模型在广泛参数范围内的平均场近似和蒙特卡罗模拟结果进行了比较。随着控制主体独立性水平的参数变化，会发生铁磁相变，该相变可以是一阶或二阶的，这取决于游说团体q的大小。这种相变的细节，例如临界点的位置、临界指数和可能的磁滞回线宽度，取决于各层的拓扑结构和其他特征，特别是取决于各层中节点的平均度，而这可以由均匀对近似直接预测。如果节点的平均度远大于q-游说团体的大小，则基于均匀对近似的数值结果与关于铁磁相变的顺序和细节的理论预测之间会取得很好的一致性。对于具有均匀厄尔多斯 - 雷尼图、随机正则图以及弱异质无标度网络形式的层的多重网络上的模型，这种一致性是定量的，而对于具有强异质无标度网络形式的层的情况，它只是定性的。如果节点的平均度较小且与q相当，均匀对近似的预测通常甚至在定性上都是错误的。
