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  • 1Departament de Física de la Matèria Condensada, Facultat de Física, <a href="https://ror.org/021018s57">Universitat de Barcelona</a>, Carrer Martí i Franquès 1, 08028 Barcelona, Spain and Institut de Nanociència i Nanotecnologia, <a href="https://ror.org/021018s57">Universitat de Barcelona</a>, Av. Joan XXIII S/N, 08028 Barcelona, Spain.

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We analyzed the Bayesian Naming Game (BNG) and β-model (β-NG) as dynamical systems. A nongeneric bifurcation at β=1/3 was found in the β-model, creating an elliptical manifold of fixed points.

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Area of Science:

  • Dynamical systems analysis
  • Statistical physics
  • Network theory

Background:

  • The β-model (β-NG) and Bayesian Naming Game (BNG) are key models for understanding language evolution and consensus formation.
  • Previous studies often assumed equal learning probabilities (pA=pB=p), limiting model applicability.

Purpose of the Study:

  • To analyze the β-model and BNG as dynamical systems.
  • To investigate bifurcations and fixed points in these models.
  • To relax the assumption of equal learning probabilities in the BNG.

Main Methods:

  • Linear stability analysis applied to the β-model's dynamical system.
  • Modeling Bayesian learning probabilities as logistic functions for the BNG.
  • Identification and characterization of fixed points and manifolds.

Main Results:

  • Demonstrated a nongeneric bifurcation in the β-model at βc=1/3.
  • Identified a one-dimensional manifold of fixed points, forming an elliptical arc, as β crosses βc.
  • Established the existence of fixed points in the BNG using logistic probability functions, without assuming pA=pB.

Conclusions:

  • The β-model exhibits complex behavior at a critical point, leading to an elliptical manifold of stable states.
  • The logistic function approach provides a more flexible framework for analyzing the Bayesian Naming Game.
  • These findings offer deeper insights into the dynamics of consensus formation and language evolution.