AB model

Let us now return to the Voter model. In the original model we had agents occupying two possible states. They chose their state simply by copying the choice made by their neighbors. Yet in most elections around the world more than two parties compete for the electoral vote. Furthermore it is hardly believable that any established supporter of any party would switch to following the opposing party over night. One way to account for these zealous supporters would be to introduce "agents with fixed state." Yet some strongly opinionated individuals do changes their beliefs, thus this would not be an ideal solution. Alternative approach was considered in [1]. In this paper a three state model is proposed, where the third state serves as intermediate stop for the agents switching between the two main states.

Model

In [1] the AB model is formulated as a model of competition between languages. As any individual can speak two languages the third state is easily justifiable - it basically means that individual know two languages. Yet, despite this difference, the underlying mechanics behind the model remains the same as in the Voter model.

Let us assume that some agents speak only A, some only B, while some speak both A and B. These agents may talk to their four neighbors. Naturally if you speak to people you eventually learn their language (understand their political views):

\begin{equation} p(A \rightarrow AB)=\frac{1}{2} x_B , \quad p(B \rightarrow AB) = \frac{1}{2} x_A . \end{equation}

Yet you will hardly forget your initial language (initial political view) over night:

\begin{equation} p( A \rightarrow B) = 0 , \quad p(B \rightarrow A)=0 . \end{equation}

Here \( x_i \) is a fraction of neighbors speaking only given language. Accounting for bilingual agents allows us to formulate law of conservation, \( x_A + x_B + x_{AB} = 1 \). The language in this model may be forgotten if agent does not use it frequently:

\begin{equation} p(AB \rightarrow B)=\frac{1}{2} (1-x_A) , \quad p(AB\rightarrow A) = \frac{1}{2} (1-x_B) . \end{equation}

Thus one just needs to randomly pick agent and change his state according to the given transition probabilities. In the applet below during one time unit all agents have a chance to switch their state once (on average as agents are picked randomly).

A similar model was earlier proposed in [2]. In [2] the authors have started from the original Voter model. This allows significantly simpler treatment in one-dimensional case, e.g, ring topology, - one can model movement of domain boundaries instead of every single agent. We have already discussed this technique in previous text "Dynamical correlated spin model" (in which we presented results of [3]).

Interactive applet

What you should check out in the applet below? First you will see a noise picture in which agents A (red), B (blue) and AB (magenta) will be intermixed (note that the grid's edges are interconnected). The initial view is determined by \( \rho_A \) and \( \rho_B \) values, which set initial density of two main agent types. Yet after short time the view will change into something reminding of Ising model. Namely uniform groups (domains) of A or B agents will form. These domains will be separated by a thin layer of AB agents. After some fast forwarding, ">>" button, you should observe a growing dominance of a single state (either A or B).

No more instructions - just try it yourself!

References

• X. Castello, V. M. Eguíluz, M. San Miguel. Ordering dynamics with two non-excluding options: bilingualism in language competition. New Journal of Physics 8: 308 (2006). doi: 10.1088/1367-2630/8/12/308. arXiv: physics/0609079 [physics.soc-ph].
• F. Vazquez, P. L. Krapivsky, S. Redner. Constrained opinion dynamics: freezing and slow evolution. Journal of Physics A 36: L61 (2003). doi: 10.1088/0305-4470/36/3/103. arXiv: cond-mat/0209445 [cond-mat.stat-mech].
• A. Kononovicius, J. Ruseckas. Stochastic dynamics of N correlated binary variables and non-extensive statistical mechanics. Physics Letters A 380: 1582-1588 (2016). doi: 10.1016/j.physleta.2016.03.006. arXiv: 1601.06968 [cond-mat.stat-mech].