Ferromagnetic and spin-glass like transition in the q-neighbor Ising model on random graphs

Abstract

The q-neighbor Ising model is investigated on homogeneous random graphs with a fraction of edges associated randomly with antiferromagnetic exchange integrals and the remaining edges with ferromagnetic ones. It is a nonequilibrium model for the opinion formation in which the agents, represented by two-state spins, change their opinions according to a Metropolis-like algorithm taking into account interactions with only a randomly chosen subset of their q neighbors. Depending on the model parameters in Monte Carlo simulations, phase diagrams are observed with first-order ferromagnetic transition, both first- and second-order ferromagnetic transitions and second-order ferromagnetic and spin-glass-like transitions as the temperature and fraction of antiferromagnetic exchange integrals are varied; in the latter case, the obtained phase diagrams qualitatively resemble those for the dilute spin-glass model. Homogeneous mean-field and pair approximations are extended to take into account the effect of the antiferromagnetic exchange interactions on the ferromagnetic phase transition in the model. For a broad range of parameters, critical temperatures for the first- or second-order ferromagnetic transition predicted by the homogeneous pair approximation show quantitative agreement with those obtained from Monte Carlo simulations; significant differences occur mainly in the vicinity of the tricritical point in which the critical lines for the second-order ferromagnetic and spin-glass-like transitions meet. Graphic abstract

Pair approximation for the q-voter models with quenched disorder on networks

“…Especially prone to this kind of error is the q-voter model with anticonformity. Similar discrepancies are reported with regard to other dynamics [28,29,32,33,35,36]. However, we have not observed such errors for the quenched counterparts of the studied models.…”

“…The pair approximation is a general technique used to study various dynamics on static [28][29][30][31][32][33][34][35][36][37][38][39][40] as well as coevolutionary networks [41][42][43][44][45]. This method has already been applied to the q-voter models with nonconformity under the annealed approach.…”

“…In particular, agents with fixed behavior classified as either conformity or nonconformity can be found in studies on the q-voter model [13,15,21], the majority-vote model [22][23][24], or the Galam model [25]. Quenched disorder may be also related with the interactions between agents [26][27][28][29]. However, there are less studies in which the annealed and quenched approaches are directly compared within the same dynamics, especially on networks.…”

Discontinuous phase transitions in the q-voter model with generalized anticonformity on random graphs

“…We decided on such a simple structure because in such a case the pair approximation (PA) should be more valid than for networks with a higher clustering coefficient 26,[34][35][36][37] . However, to validate the analytical results, we conduct also Monte Carlo simulations, because it was reported recently that PA can give invalid results even on ER graphs if the mean degree of nodes is small and comparable with the size of the influence group q 10,29,38 . In such a case, the results can be wrong not only quantitatively, but even qualitatively, indicating discontinuous phase transitions, which are not observed in the computer simulations.…”