  Independence and anticonformity are two types of social behaviors known in
social psychology literature and the most studied parameters in the opinion
dynamics model. These parameters are responsible for continuous (second-order)
and discontinuous (first-order) phase transition phenomena. Here, we
investigate the majority rule model in which the agents adopt independence and
anticonformity behaviors. We define the model on several types of graphs:
complete graph, two-dimensional (2D) square lattice, and one-dimensional (1D)
chain. By defining $p$ as a probability of independence (or anticonformity), we
observe the model on the complete graph undergoes a continuous phase transition
where the critical points are $p_c \approx 0.334$ ($p_c\approx 0.667$) for the
model with independent (anticonformist) agents. On the 2D square lattice, the
model also undergoes a continuous phase transition with critical points at $p_c
\approx 0.0608$ ($p_c \approx 0.4035$) for the model with independent
(anticonformist) agents. On the 1D chain, there is no phase transition either
with independence or anticonformity. Furthermore, with the aid of finite-size
scaling analysis, we obtain the same sets of critical exponents for both models
involving independent and anticonformist agents on the complete graph.
Therefore they are identical to the mean-field Ising model. However, in the
case of the 2D square lattice, the models with independent and anticonformist
agents have different sets of critical exponents and are not identical to the
2D Ising model. Our work implies that the existence of independence behavior in
a society makes it more challenging to achieve consensus compared to the same
society with anticonformists.
