In October 2021, South American Institute of Fundamental Research of International Centre for Theoretical Physics hosted a workshop on Sociophysics. If you're looking to get started in doing opinion dynamics, this recording of the talks by the two leading researchers in the field, Katarzyna Sznajd-Weron and Maxi San Miguel, is a great place to begin. If you are interested, complete program and recordings of other talks are available on dedicated site

With this video we end this academic year and wish you relaxing summer holidays.

In a recent post, we have discussed how to obtain the stationary variance of the AR(2) process using Yule-Walker equations. While intuitively, it is trivial to see that the poll-delayed voter model is an AR(2) process, showing this formally is a bit more involved. Let us delve into this question further.

It is hard (or impossible) to directly obtain the analytical expression for the stationary distribution of the poll-delay voter model. But we can look at various possible approximations, with the Beta distribution being the prime suspect. To fit the Beta distribution (or any other two-parameter distribution), we need to know two stationary moments of the model. Deriving the stationary mean is a trivial problem, while deriving the stationary variance is more involved.

In this post, let us use the Yule-Walker equations to obtain the expression for stationary variance of AR(2) process.

Important thing to notice about the poll-delayed voter model is that it is a second order auto-regressive process. In [1] we have used this observation to derive an analytical expression for the stationary variance for the model. To understand how this result came to be, the following video by ritvikmath might prove useful.

References

• A. Kononovicius, R. Astrauskas, M. Radavičius, F. Ivanauskas. Delayed interactions in the noisy voter model through the periodic polling mechanism. Physica A 652: 130062 (2024). doi: 10.1016/j.physa.2024.130062. arXiv:2403.10277 [physics.soc-ph].

In the last few posts, I have introduced you to two variations of the voter model enhanced with polling mechanism [1]. These two variations differed in whether the announcement of poll outcome is delayed or not. This time, we will explore the distributions obtained from these models.