Voter model

In 1973 two British scientists proposed an elementary model of spatial conflict between two distinct species [1]. The results of this model were heavily dependent on the dimensionality of the topology on which modeling occurred, which was rather interesting phenomenon at that time. In contemporary science this model is rather popular, thought now it is known as the voter model. This text discusses the simplest version of the model.

Essential rules behind the model

We start with grid, which is fully filled with agents of random types. In our app we use only two types, but in general case one could use more. Next during each time step random agent chooses his random neighbor and copies his "type" (imitates his behavior). Most of other voter models differ in stochastic rule, which is used to select random neighbor. In our app all immediate neighbors are chosen with equal probability.

Why this model is referred to as the voter model? Most of the more sophisticated applications of this model consider opinion dynamics. It has becomes one of the iconic models of opinion dynamics in sociology. It actually doesn't matter if you model human or animals. It actually doesn't matter if competition is between different opinions or between different species.

HTML5 app

In this app you can choose dimensionality of the topology - "1D" stands for the one dimensional case (topology - ring), while "2D" stands for the two dimensional case (topology - torus). You can also set the probability that agent will start being of "red" type, \( p_r \). On the right hand side of the app you will see a graph of the mean opinion. While calculating mean opinion it is assumed that "red" agents have opinion index \( 1 \), while blue agents have opinion index \( -1 \).

References

• P. Clifford, A. Sudbury. A model for spatial conflict. Biometrika 60: 581-588 (1973). doi: 10.1093/biomet/60.3.581.