# American Institute of Mathematical Sciences

doi: 10.3934/nhm.2021006

## Opinion fitness and convergence to consensus in homogeneous and heterogeneous populations

 1 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, C/ Tarfia s/n, and Instituto de Matemáticas Antonio de Castro Brzezicki, Edificio Celestino Mutis, Avda. de la Reina Mercedes s/n, Universidad de Sevilla, Campus de Reina Mercedes, (41012) Sevilla, Spain 2 IMAS UBA-CONICET and Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Av Cantilo s/n, Ciudad Universitaria, (1428) Buenos Aires, Argentina

Received  August 2020 Revised  November 2020 Published  February 2021

Fund Project: This work was partially supported by Universidad de Buenos Aires under grants 20020150100154BA and 20020130100283BA, by ANPCyT PICT2012 0153 and PICT2014-1771, CONICET (Argentina) PIP 11220150100032CO and 5478/1438. J.P. Pinasco and N. Saintier are members of CONICET, Argentina. M. Pérez-Llanos thanks to Junta de Andalucía FQM-131, Spain

In this work we study the formation of consensus in homogeneous and heterogeneous populations, and the effect of attractiveness or fitness of the opinions. We derive the corresponding kinetic equations, analyze the long time behavior of their solutions, and characterize the consensus opinion.

Citation: Mayte Pérez-Llanos, Juan Pablo Pinasco, Nicolas Saintier. Opinion fitness and convergence to consensus in homogeneous and heterogeneous populations. Networks & Heterogeneous Media, doi: 10.3934/nhm.2021006
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##### References:
Evolution of the opinion of 10 agents (blue) from a homogeneous population of $N = 1000$ agents interacting according the interaction rule (16) studied in this paper (left) or the interaction rule (22) considered in [2,38] (right). The red dashed line indicates the theoretical limit opinion in both cases ($m_\infty\approx -0.35$ for interaction rule (16), (left), and $\tilde m_\infty\approx 0.41$ for interaction rule (22), (right)) The early evolution is shown in inset
Evolution of $\ln(\max\,w-\min\,w)$, where the $\max$ and $\min$ are taken on the support of the distribution of opinions, for interaction rule (16) studied in this paper (left), and interaction rule (22) considered in [2,38] (right)
Evolution of the opinion of 10 agents (blue) from a population of $N = 1000$ agents with $\alpha_0 = 2\%$ (left) and $\alpha_0 = 60\%$ (right) stubborn agents. The red dashed line indicates the theoretical limit opinion $m_\infty = 1/2$, and the blue dotted lines the opinion of the stubborn agents (half with opinion $1/4$ and the other half with opinion $3/4$). The early evolution is shown in inset
Evolution of the distribution of $(w,q)$ among the non-stubborn population during one simulation ($w$ in the horizontal axis and $q$ in the vertical axis) with $\lambda(w) = (w-0.5)^2 + \varepsilon$, $\varepsilon = 0.01$. From left to right and top to bottom, figures show the distribution of $(w,q)$ at time $1,500,1000,1500,3000,5000,10000,15000,30000$
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