The fractal geometry of opinion formation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918312743534592 |
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| author | Cao, Fei Cortez, Roberto |
| author_facet | Cao, Fei Cortez, Roberto |
| contents | In this manuscript, we introduce and study a variant of the agent-based opinion dynamics proposed in a recent work [8], 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 $μ_+ \in (0,1]$ towards $+1$, or by a proportion $μ_- \in (0,1]$ 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 [23,44]. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_21023 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The fractal geometry of opinion formation Cao, Fei Cortez, Roberto Probability In this manuscript, we introduce and study a variant of the agent-based opinion dynamics proposed in a recent work [8], 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 $μ_+ \in (0,1]$ towards $+1$, or by a proportion $μ_- \in (0,1]$ 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 [23,44]. 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. |
| title | The fractal geometry of opinion formation |
| topic | Probability |
| url | https://arxiv.org/abs/2601.21023 |