A maximum entropy model for opinions in social groups

We study how the opinions of a group of individuals determine their spatial distribution and connectivity, through an agent-based model. The interaction between agents is described by a Potts-like Hamiltonian in which agents are allowed to move freely without an underlying lattice (the average network topology connecting them is determined from the parameters). This kind of model was derived using maximum entropy statistical inference under fixed expectation values of certain probabilities that (we propose) are relevant to social organization. Control parameters emerge as Lagrange multipliers of the maximum entropy problem, and they can be associated with the level of consequence between the personal beliefs and external opinions, and the tendency to socialize with peers of similar or opposing views. These parameters define a phase diagram for the social system, which we studied using Monte Carlo Metropolis simulations. Our model presents both first and second-order phase transitions, depending on the ratio between the internal consequence and the interaction with others. We have found a critical value for the level of internal consequence, below which the personal beliefs of the agents seem to be irrelevant.

💡 Research Summary

The paper presents a statistical‑mechanics framework for modeling opinion dynamics in social groups, derived from the principle of maximum entropy. Each of the N agents possesses three attributes: a continuous spatial position in two dimensions, an internal belief (B_i) (an integer between 1 and Q that represents the opinion the agent truly holds), and an external expressed opinion (S_i) (also an integer between 1 and Q, which may differ from (B_i) under social pressure). The authors assume that the internal beliefs are fixed for the duration of the study, while the external opinions and positions can evolve.

Three observable quantities are taken as constraints in the maximum‑entropy inference: (i) the probability that an agent’s external opinion matches its internal belief, denoted (P_C); (ii) the joint probability that two agents are both spatially close (distance (r_{ij}<R_c)) and share the same external opinion, denoted (P_J); and (iii) the probability that two agents are within the interaction radius, denoted (P_R). By introducing Lagrange multipliers (\lambda_C, \lambda_J,) and (\lambda_R) for these constraints and maximizing Shannon entropy, the authors obtain a canonical distribution \