Three-State Majority-Vote Model on Relative Neighborhood Graph

Agent-based models and network science are crucial for investigating social, biological and economic emergent behavior within human society. The complex structure of connections among individuals is an essential characteristic of social systems, supporting the collective and interacting critical phenomena. Using the Relative Neighborhood Graphs (RNG), we explore opinion formation dynamics and consensus break under the influence of proximity effects in social interactions. We distribute the network nodes in a regular square lattice with a placement randomness factor s that alters the spatial positions of the nodes and build the RNG to capture local proximity relationships efficiently. We investigate how the proximity network structure affects the opinion dynamics of the three-state majority-vote model, in which an agent's opinion may assume the values +1, 0, and -1, and a social temperature q induces a level of disagreement within the system. Using Monte Carlo simulations, we compute essential quantities like opinionization, opinion susceptibility, and the Binder fourth-order cumulant for different levels of displacement s. Our findings show that the system exhibits a continuous phase transition influenced by the disorder parameter s and social temperature q.