Discrete-Time Hegselmann-Krause Model for a Leader-Follower Social Network

The Hegselmann-Krause model (H-K model) is popular for studying opinion dynamics in social networks. All agents in a basic H-K model have identical roles. However, in many practical multi-agent opinion systems there are special agents, called leaders, which can influence their neighbors but cannot be influenced by them. There is relatively little research reported on this scenario. In this paper, we incorporate this new leader-follower feature into an H-K model. In particular, we study the consensus problem of the new leader-follower H-K model in two cases. First, when the leader's opinion is fixed (that is, it does not change over time), we give a necessary and sufficient condition for achieving a consensus for these systems and discuss whether the time of convergence is finite or infinite. Second, when the leader's opinion changes according to a linear function, we derive an upper bound on the drifting rate of the leader for the system to maintain an epsilon -> profile. The epsilon - profile is an important characteristic of an H-K model. We also establish a sufficient condition on the basis of the leader's initial state and drifting rate for the system to achieve an epsilon - profile. Finally, we show that a leader can actively control the system to a consensus if nonlinear controls are allowed. One interesting discovery is that a leader aiming to drive all followers to a high value consensus should start with a low enough opinion state and increase the state value at a slow pace.