On the synchronization of coupled random walks and applications to social systems

Abstract: Some political strategies to win elections over the last years were based heavily on fomenting general distrust in information institutions and favoring distrustful sources. The misinformation pandemic has the straightforward consequence that people do not believe any information unless it is compatible with their own beliefs. We present a simple model to study the emergence of consensus in opinion pools of uncertain agents that trust (couple to) their neighbors (information sources) with strength K. We focus the studies on regular lattices and linear coupling. Depending on the coupling constant, K, and the propensity to choose an opinion (the probability to manifest a given opinion in solitude), p_0, we get regions where consensus is surely reached even in infinity systems (K greater than or equal to K_c), regions where not-consensus is the only steady state (K lesser than K_c), and a region where consensus on any opinion is transitory with each agent presenting periods of strong oscillation of opinions before changing polarization (p_0 equal to p_c and K greater than or equal K_c. The first model in this last region presents transition probabilities identical to the voter model (p_0 equal to p_c and K equal to K_c). Different upbringings, exposition to education biased to a single political view, and previous coexistence with opinion polarized people can change the opinion of agents in isolation. We model such characteristics with heterogeneous populations (p^i_0 not equal to p^j_0 for some pairs of nodes i,j). Such systems present regions where the coexistence of local consensus (bulk-stable clusters of like-minded opinions), weak consensus (bulk-unstable temporary clusters where contrary opinions emerge inside the cluster), and distrust (random orientations that do not form clusters) are possible.