Practical consensus in bounded confidence opinion dynamics

Opinion dynamics expressed by the bounded confidence discrete-time heterogeneous Hegselmann–Krause model is considered. A policy for the adaptation of the agents confidence thresholds based on heterophily, maximum number of neighbors and non-influencing similarity interval is proposed. The policy leads to the introduction of the concepts of practical clustering and practical consensus. Several properties of the agents dynamic behaviors are proved by exploiting the roles of the agents having at each time-step the maximum and the minimum opinions. The convergence in finite time to (a maximum number of) practical clusters and, for sufficiently large threshold bounds, the convergence to a practical consensus are proved. Sufficient conditions for reaching a practical consensus around a stubborn are derived too. Numerical simulations verify the theoretical results.