Fractal Opinions among Interacting Agents

We investigate an opinion model consisting of a large group of interacting agents, whose opinions are represented as numbers in [-1,1]. At each update time, two random agents are selected, and the opinion of the first agent is updated based on the opinion of the second (the ``persuader""). We derive rigorously the mean-field kinetic equation describing the large population limit of the model, and we provide several quantitative results establishing convergence to the unique equilibrium distribution. Surprisingly, in some range of the model parameters, the support of the equilibrium distribution exhibits a fractal structure, linking the mean-field description of our opinion dynamics to the concept of Bernoulli convolutions studied extensively in the fractal geometry literature [P. Erd\" os, Amer. J. Math., 61 (1939), pp. 974-976], [P. P. Varj\' u, European Congress of Mathematics, European Mathematical Society (EMS), Z\" urich, 2018, pp. 847-867]. This provides a new mathematical description for the so-called opinion fragmentation phenomenon.