Extending the Biswas--Chatterjee--Sen model with nonconformists and inflexibles

Abstract

Originally, the Biswas--Chatterjee--Sen model exhibits an order/disorder phase transition for a sufficiently large number of negative interactions among actors. In this paper, the model is extended by the nonconformists and inflexibles. Nonconformists are actors who do not follow the original model rules, but in different ways do something opposite. We introduce inflexibles as actors who does not change their opinions. Both discrete and continuous opinions are considered. With direct Monte Carlo simulations and mean-field calculations, we check the influence of fractions of nonconformists and inflexibles on mean opinion in the system. With the mean-field calculations we identify ranges of fractions of nonconformists where ordered phase of the system is available. The results of the mean-field calculations perfectly match the results of the Monte Carlo simulations. We consider inflexibles adhered: (i) to extreme opinions; (ii) to specific opinions and (iii) chosen independently of their initial opinion. For inflexibles adhered to specific and extreme opinions they play a role of effective bias suppressing disorder phase in the system. The qualitative results of introducing nonconformists (inflexibles) in various ways (discrete/continuous opinions and annealed/quenched disorder) are roughly the same. However, for the model extended by inflexibles, we can observe a systematic shift of the mean order parameter to its higher values for quenched disorder compared with annealed disorder.

Figures (8)

• Figure 1: Mean order parameter $\langle\{O\}\rangle$ for various fractions $c$ of nonconformists with discrete opinions modeled according to \ref{['eq:model_c1']} (upper panel) and \ref{['eq:model_c2']} (lower panel). The annealed disorder is assumed. $N=1024$, $T=10^3$, $R=10^3$, $\tau=0.2T$
• Figure 2: Mean order parameter $\langle\{O\}\rangle$ for various fractions $z$ of inflexibles with discrete opinions. The inflexibles are adhered to opinion $+1$. The quenched (upper panel) or annealed (lower panel) disorder is assumed. $N=1024$, $T=10^3$, $R=10^3$, $\tau=0.2T$
• Figure 3: Mean order parameter $\langle\{O\}\rangle$ for various fractions $c$ of nonconformists for continuous opinions. The upper panel corresponds to nonconformists introduced according to \ref{['eq:model_c1']}, while the lower shows results for implementing rule given by \ref{['eq:model_c2']}. The quenched disorder is assumed. $N=1024$, $T=10^3$, $R=10^3$, $\tau=0.2T$
• Figure 4: Mean order parameter $\langle\{O\}\rangle$ for various fractions $z$ of inflexibles for continuous opinions. The inflexibles are adhered to opinion $+1$. Only the quenched disorder may be considered. $N=1024$, $T=10^3$, $R=10^3$, $\tau=0.2T$
• Figure 5: Mean order parameter $\langle\{O\}\rangle$ for various fractions $c$ of nonconformists with discrete opinions modeled according to \ref{['eq:model_c1']}. The solid line represent the closed form expression of the order parameter given in \ref{['ordered fixed points']} while the points denote the simulation results. The analytical prediction shows excellent agreement with the simulation data. All simulations were done for the system size $N=1024$