Local dynamics vs. social mechanisms:: A unifying frame

We present a general sequential probabilistic frame, which extends a series of earlier opinion dynamics models. In addition, it orders and classifies all of the existing two-state spin systems. The scheme operates via local updates where a majority rule is applied differently in each possible configuration of a local group. It is weighted by a local probability which is a function of the local value of the order parameter, i.e., the majority-to-minority ratio. The system is thus driven from one equilibrium state into another equilibrium state till no collective change occurs. A phase diagram can thus be constructed. It has two phases, one where the collective opinion ends up broken along one opinion, and another with an even coexistence of both opinions. Two different regimes, monotonic and dampened oscillatory, are found for the coexistence phase. At the phase transition local probabilities conserve the density of opinions and reproduce the collective dynamics of the Voter model. The essential behavior of all existing discrete two-state models (Galam, Sznajd, Ochrombel, Stauffer, Krapivsky-Redner, Mobilia-Redner, Behera-Schweitzer, Slanina-Lavicka, Sanchez...) is recovered and found to depart from each other only in the value of their local probabilities. Corresponding simulations are discussed. It is concluded that one should not judge from the above model results the validity of their respective psycho-social assumptions.