Stationary distribution of the noisy voter model with supportive interactions

In the last few posts we have discussed voter model with supportive interactions. In most cases the support is strong and drives even non-extensive model, which is described by a broad stationary distribution in the absence of support, to a stationary point. Yet there are cases when such driving is not overly strong and some stochastic behavior is retained. In this post we present an app, which also allows you to examine stationary PDF of the model, where support suppresses recruitment.

If \( q < \min(\sigma_0, \sigma_1) \), then the model has stationary Beta distribution:

\begin{equation} x\sim\mathcal{B}e\left(\frac{\sigma_{1}-qN^{1-\beta}}{h+qN^{-\beta}},\frac{\sigma_{0}-qN^{1-\beta}}{h+qN^{-\beta}}\right). \end{equation}

If \( q > \min(\sigma_0, \sigma_1) \), it is not clear whether the model has stationary distribution. Quite likely, for the smaller \( q \), the model exhibits weak ergodicity breaking. Detailed understanding of this phase of the model is still forthcoming.

Interactive app below is almost equivalent to the one used in the previous post. The only difference is that PDF plot in the log-linear scales is also shown. Simulated PDF is shown as grey dots, while Beta PDF is shown if we know that stationary distribution exists.

Acknowledgment. This post was written while reviewing literature relevant to the planned activities in postdoctoral fellowship ''Physical modeling of order-book and opinion dynamics'' (09.3.3-LMT-K-712-02-0026) project. The fellowship is funded by the European Social Fund under the No 09.3.3-LMT-K-712 ''Development of Competences of Scientists, other Researchers and Students through Practical Research Activities'' measure.