Abstract:In this manuscript, we introduce and study a variant of the agent-based opinion dynamics proposed in a recent work [9], within the framework of an interacting multi-agent system, where agents are assumed to interact with each other and update their opinions after each pairwise encounter. Specifically, our opinion model involves a large crowd of $N$ indistinguishable agents, each characterized by an opinion value ranging within the interval $[-1,1]$. At each update time, two agents are picked uniformly at random and the opinion of one agent will either shift by a proportion $\mu \in (0,1]$ towards $+1$, or by a proportion $\lambda \in (0,1]$ towards $-1$, with probabilities depending on the other agent's opinion. We rigorously derive the mean-field limit PDE that governs the large-population limit of the agent-based model and present several quantitative results demonstrating convergence to the unique equilibrium distribution. Remarkably, for a suitable choice of model parameters, the long-term equilibrium opinion profile displays a striking self-similar structure that generalizes the celebrated Bernoulli convolution, a topic extensively studied in the context of fractal geometry [24,51]. These findings also enhance our understanding of the opinion fragmentation phenomenon and may provide valuable insights for the development of more sophisticated models in future research.
From: Roberto Cortez [view email]
[v1]
Wed, 28 Jan 2026 20:34:30 UTC (515 KB)
[v2]
Mon, 27 Jul 2026 19:15:41 UTC (559 KB)
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