  We explore the impact of social noise, characterized by nonconformist
behavior, on the phase transition within the framework of the majority-rule
model. The order-disorder transition can reflect the consensus-polarization
state in a social context. This study covers various network topologies,
including complete graphs, two-dimensional (2-D) square lattices,
three-dimensional (3-D) square lattices, and heterogeneous or complex networks
such as Watts-Strogatz (W-S), Barab\'asi-Albert (B-A), and Erdos-R\'enyi (E-R)
networks, as well as their combinations. Social behavior is represented by the
parameter $p$, which indicates the probability of agents exhibiting
nonconformist behavior. Our results show that the model exhibits a continuous
phase transition across all networks. Through finite-size scaling analysis and
evaluation of critical exponents, our results suggest that the model falls into
the same universality class as the Ising model.
