Opinion Dynamics Control and Consensus in a Social Network

We consider a game-theoretic model of the influence of players on the dynamics of opinions and the consensus achieved in a social network. The control problem is to maintain the opinions of all participants in the vicinity of a predetermined value. If there are several players, then these target values may differ. The considered dynamic game belongs to the class of linear-quadratic games in discrete time. Optimal control and equilibrium are found using the Bellman equation. The solution is reached in closed form. It is shown that in the model with one player, a controlled consensus is achieved in the social network. In the model with two players, it is shown that although there is no consensus in the social network, the equilibrium is completely determined by the mean value of the opinions of all participants, which converges to a certain value. The results of numerical modeling for a social network with one and two players are presented.