Statistical complexity as a diagnostic of critical behavior in a 2D Sznajd model

We investigate the dynamics of a two-dimensional Sznajd model in which a third ideological position emerges spontaneously from local interactions between agents. Starting from random initial conditions, we observe that this emergent position arises solely as a result of the interaction dynamics, and not from predefined agent states. We analyze the system's behavior as a function of social apathy, modeled as the fraction of nonparticipating agents, and identify a critical apathy threshold that separates distinct regimes of opinion dominance. To quantify the interplay between disorder and structure in the system, we employ the complexity-entropy plane. This analysis reveals that the system is robust to asymmetries in initial conditions; its macroscopic behavior remains qualitatively unchanged. Furthermore, we find that statistical complexity reaches a maximum near the transition region, highlighting the emergence of rich collective dynamics characterized by abrupt topological changes parametrized by apathy.