Leader-Follower Dynamics

Abstract

The original Leader-Follower model, proposed in \cite{zhao2018understanding}, categorizes agents with opinions in $[-1,1]$ into a follower group, a leader group with a positive target opinion in $[0,1]$, and a leader group with a negative target opinion in $[-1,0]$. Leaders maintain a constant attraction to their target, blending it with the average opinion of their group neighbors at each update. Followers, on the other hand, have a constant attraction to the average opinion of their leader group's opinion neighbors, also integrating it with their group neighbors' average opinion. This model was numerically studied in \cite{zhao2018understanding}. This paper extends the Leader-Follower model to include a social relationship, variable degrees over time, high-dimensional opinions, and a flexible number of leader groups. We theoretically investigate conditions for asymptotic stability or consensus, particularly in scenarios where a few leaders can dominate the entire population.

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5
• proof
• Lemma 6
• proof
• proof : Proof of Theorem \ref{['Thm:consensus equal to target']}
• Lemma 7