Leader–follower dynamics: stability and consensus in a socially structured population

Hsin-Lun Li; Hsin-Lun Li

doi:10.3934/math.2025169

AIMS Mathematics

2025, Volume 10, Issue 2: 3652-3671. doi: 10.3934/math.2025169

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Leader–follower dynamics: stability and consensus in a socially structured population

Hsin-Lun Li ^,

National Sun Yat-sen University, Kaohsiung 804, Taiwan

Received: 26 December 2024 Revised: 13 February 2025 Accepted: 17 February 2025 Published: 25 February 2025
MSC : 91C20, 91D25, 91D30, 93D50, 94C15

The original leader–follower model 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.
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.
- social network,
- leader–follower dynamics,
- consensus,
- Hegselmann–Krause dynamics,
- averaging dynamics
Citation: Hsin-Lun Li. Leader–follower dynamics: stability and consensus in a socially structured population[J]. AIMS Mathematics, 2025, 10(2): 3652-3671. doi: 10.3934/math.2025169

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Abstract

The original leader–follower model 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.

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.

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