| Abstract: |
| We study time-delayed variants of the Hegselmann-Krause opinion formation model. In particular, we focus on a model featuring a small group of leaders and a large group of non-leaders. In our model, leaders influence all agents but only interact among themselves, while non-leaders update their opinions via interactions with both their peers and the leaders, with time delays accounting for communication and decision-making lags. We prove that the system achieves consensus with an exponential decay rate and establish uniform $l_\infty$-stability with exponentially decaying transients. Furthermore, we analyze the mean-field limit in two regimes: (i) with a fixed number of leaders and an infinite number of non-leaders, and (ii) with both populations tending to infinity, obtaining existence, uniqueness, and exponential decay estimates for the corresponding macroscopic models. |
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