Krawiecki A, Gradowski T
Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL-00-662 Warsaw, Poland.
Phys Rev E. 2023 Jul;108(1-1):014307. doi: 10.1103/PhysRevE.108.014307.
The q-neighbor Ising model for the opinion formation on multiplex networks with two layers in the form of random graphs (duplex networks), the partial overlap of nodes, and LOCAL&AND spin update rule was investigated by means of the pair approximation and approximate master equations as well as Monte Carlo simulations. Both analytic and numerical results show that for different fixed sizes of the q-neighborhood and finite mean degrees of nodes within the layers the model exhibits qualitatively similar critical behavior as the analogous model on multiplex networks with layers in the form of complete graphs. However, as the mean degree of nodes is decreased the discontinuous ferromagnetic transition, the tricritical point separating it from the continuous transition, and the possible coexistence of the paramagnetic and ferromagnetic phases at zero temperature occur for smaller relative sizes of the overlap. Predictions of the simple homogeneous pair approximation concerning the critical behavior of the model under study show good qualitative agreement with numerical results; predictions based on the approximate master equations are usually quantitatively more accurate but yet not exact. Two versions of the heterogeneous pair approximation are also derived for the model under study, which, surprisingly, yield predictions only marginally different or even identical to those of the simple homogeneous pair approximation. In general, predictions of all approximations show better agreement with the results of Monte Carlo simulations in the case of continuous than discontinuous ferromagnetic transition.
通过对近似主方程以及蒙特卡罗模拟的对近似方法，研究了具有两层随机图（双链网络）形式、节点部分重叠且采用LOCAL&AND自旋更新规则的多路复用网络上意见形成的q邻域伊辛模型。解析和数值结果均表明，对于不同固定大小的q邻域以及层内节点的有限平均度，该模型表现出与具有完全图形式层的多路复用网络上的类似模型在定性上相似的临界行为。然而，随着节点平均度的降低，对于较小的重叠相对大小，会出现不连续的铁磁转变、将其与连续转变分开的三临界点以及在零温度下顺磁相和铁磁相可能的共存。关于所研究模型临界行为的简单均匀对近似预测与数值结果显示出良好的定性一致性；基于近似主方程的预测通常在定量上更准确，但仍不精确。还为所研究的模型推导了两种非均匀对近似版本，令人惊讶的是，它们产生的预测与简单均匀对近似的预测仅略有不同甚至完全相同。一般而言，在连续铁磁转变的情况下，所有近似的预测与蒙特卡罗模拟结果的一致性比不连续铁磁转变的情况更好。
