Generalized Naming Game and Bayesian Naming Game as Dynamical Systems

We study the $\beta$-model ($\beta$-NG) and the Bayesian Naming Game (BNG) as dynamical systems. By applying linear stability analysis to the dynamical system associated with the $\beta$-model, we demonstrate the existence of a non-generic bifurcation with a bifurcation point $\beta_c = 1/3$. As $\beta$ passes through $\beta_c$, the stability of isolated fixed points changes, giving rise to a one-dimensional manifold of fixed points. Notably, this attracting invariant manifold forms an arc of an ellipse. In the context of the BNG, we propose modeling the Bayesian learning probabilities $p_A$ and $p_B$ as logistic functions. This modeling approach allows us to establish the existence of fixed points without relying on the overly strong assumption that $p_A = p_B = p$, where $p$ is a constant.