What is a leader of opinion formation in bounded confidence models?

Abstract: Taking a decision in democratic social groups is based on the opinion of the majority or on the consensus. So, the study of opinion dynamics is of great interest in analyzing social phenomena. Among the different models of opinion dynamics, bounded confidence models have been studied in different contexts and shown interesting dynamics [1-3]. In [E. Kurmyshev, H.A. Ju\'arez, and R.A. Gonz\'alez-Silva, Phys. A 390, 16 (2011)] we proposed a new bounded confidence model and studied the self-formation of opinion in heterogeneous societies composed by agents of two psychological types, concord (C-) and partial antagonism (PA-) agents. In this work we study the influence of "leaders" on the clustering of opinions. Mixed C/PA-societies along with the pure C- and PA-society are studied. The influence of the leader's connectivity in the network, his toughness or tolerance and his opinion on the opinion dynamics is studied as a function of the initial opinion uncertainty (tolerance) of the population. Numerical results obtained with leaders at low, high and average tolerance show complex bifurcation patterns of the group opinion; a decrease or even the total lost of control of the leader over the society is observed in different intervals of tolerance of agents in the case of C/PA-societies. We found that in the C-society a leader showing high opinion tolerance has more control over the population. In the PA-society a leader changes the bifurcation pattern of group opinion in a drastic and unexpected way, contrary to the common sense, and generates stronger polarization in the opposite opinion groups; the connectivity of the leader is an important factor that usually improves the adhesion of agents to the leader's opinion. A low tolerance (authoritarian) leader has greater control over a PA-society than that of a high tolerance (democratic) one; the opposite result is obtained in the C-society.