The qq-neighbor Ising model on multiplex networks with partial overlap of nodes

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.Comment: 21 pages, 5 figure

QQ-voter model with independence on signed random graphs: homogeneous approximations

The $q$-voter model with independence is generalized to signed random graphs and studied by means of Monte Carlo simulations and theoretically using the mean field approximation and different forms of the pair approximation. In the signed network with quenched disorder, positive and negative signs associated randomly with the links correspond to reinforcing and antagonistic interactions, promoting, respectively, the same or opposite orientations of two-state spins representing agents' opinions; otherwise, the opinions are called mismatched. With probability $1-p$, the agents change their opinions if the opinions of all members of a randomly selected $q$-neighborhood are mismatched, and with probability $p$, they choose an opinion randomly. The model on networks with finite mean degree $\langle k \rangle$ and fixed fraction of the antagonistic interactions $r$ exhibits ferromagnetic transition with varying the independence parameter $p$, which can be first- or second-order, depending on $q$ and $r$, and disappears for large $r$. Besides, numerical evidence is provided for the occurrence of the spin-glass-like transition for large $r$. The order and critical lines for the ferromagnetic transition on the $p$ vs. $r$ phase diagram obtained in Monte Carlo simulations are reproduced qualitatively by the mean field approximation. Within the range of applicability of the pair approximation, for the model with $\langle k \rangle$ finite but $\langle k \rangle \gg q$, predictions of the homogeneous pair approximation concerning the ferromagnetic transition show much better quantitative agreement with numerical results for small $r$ but fail for larger $r$. A more advanced signed homogeneous pair approximation is formulated which distinguishes between classes of active links with a given sign connecting nodes occupied by agents with mismatched opinions...Comment: 20 pages, 10 figure

Spin-glass-like transition in the majority-vote model with anticonformists

Majority-vote model on scale-free networks and random graphs is investigated in which a randomly chosen fraction p of agents (called anticonformists) follows an antiferromagnetic update rule, i.e., they assume, with probability governed by a parameter q (0 < q < 1∕2), the opinion opposite to that of the majority of their neighbors, while the remaining 1 − p fraction of agents (conformists) follows the usual ferromagnetic update rule assuming, with probability governed by the same parameter q, the opinion in accordance with that of the majority of their neighbors. For p = 1 it is shown by Monte Carlo simulations and using the Binder cumulants method that for decreasing q the model undergoes second-order phase transition from a disordered (paramagnetic) state to a spin-glass-like state, characterized by a non-zero value of the spin-glass order parameter measuring the overlap of agents’ opinions in two replicas of the system, and simultaneously by the magnetization close to zero. In the case of the model on scale-free networks the critical value of the parameter q weakly depends on the details of the degree distribution. As p is decreased, the critical value of q falls quickly to zero and only the disordered phase is observed. On the other hand, for p close to zero for decreasing q the usual ferromagnetic transition is observed