Majority-vote model

Couple of years ago we have discussed one of the most prominent models in opinion dynamics - voter model. This time we consider a generalization of the voter model, which draws inspiration from the Ising model. This generalization, known as majority-vote model [1, 2], adds thermal noise and hence the average opinion, \( M \), no longer converges to a fixed point if the amount of noise is just right.

Majority-vote model

Although the model can be run on any complex network, let us implement it on a grid with periodic boundary conditions (basically our grid geometry is torus).

Each cell on this grid corresponds to single agent. Each agent has opinion on some topic and the opinion can be binary (though it seems that the model can be generalized for non-binary opinions [3]). For the sake of simplicity let us quantify the possible alternatives as \( +1 \) (red color in our app) and \( -1 \) (blue color in our app).

During each time step a random agent is chosen. The selected agent checks whether his opinion is "aligned" with his neighbors cumulative opinion. In other words the agent checks if he agrees with the majority of his neighbors. If he agrees with the majority, then his willingness to change the opinion will be low. Otherwise the agent will be more inclined to switch his opinion. Let us say that the agent will change his opinion with probability \( q \) if he finds that he agrees with the local majority, while otherwise, if he disagrees with the local majority, he will "flip" with probability \( 1-q \). If neither of the binary opinions has majority, then the agent will flip with probability \( 0.5 \).

These forms of probabilities likely come from a specific interpretation of the Ising model. In which the orientation of the selected spin is determined by coming into contact with some heatbath. Yet in the majority-vote model it is not necessary to ensure that flipping probabilities sum to unity. It would suffice to assume that probability to flip "in-majority" is a lot smaller than probability to flip "out-of-majority". But we keep the original notation, because it is commonly used in the literature as well as to keep the model simpler (a single parameter instead of two).

To keep matters simple in this implementation of the majority-vote model we consider only von Neumann neighborhood of radius 1 (namely each agent has only 4 neighbors). In general one could consider broader radius or different types of neighborhoods, but this selection is one of the simplest and the most common in the literature.

So for the majority-vote model to work nicely we simply have to require that \( q \ll 1 \). Earlier research suggests [2] that especially interesting behavior should be observed for \( q \approx 0.075 \), because this is a critical point for continuous transition. It is also reported that this model possesses same critical exponents as the Ising model.

HTML5 app

Below you can find HTML5 app using which you can check your intuitions. Here we also provide some of our intuitions.

Obviously if \( q=0 \), then the majority-vote model becomes extremely similar to the voter model. Namely average opinion will converge to some fixed point (though not necessarily \( -1 \) or \( 1 \)).

For small yet positive \( q \) the model will start to oscillate between the \( -1 \) and \( 1 \). Though the switching from one extreme to the other could take long time. As \( q \) approaches the critical point the extremes will become less probable than broad range of medium values. For even larger \( q \) the average opinion will start to converge to \( 0 \) (approximately equal number of agents having the opposite opinions). Similar transition could be observed in the Kirman's model, which suggests an interesting research direction for us in the future.

Note that the model app provides another parameter \( p \), which sets the initial fraction of red agents (ones with \( +1 \) opinion). It is only used to generate initial configuration of the grid.

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.

References

• T. M. Liggett. Interacting Particle Systems. Springer, New York, 1985.
• M. J. de Oliveira. Isotropic majority-vote model on a square lattice. Journal of Statistical Physics 66: 273 (1992).
• F. W. S. Lima. Three-state majority-vote model on square lattice. Physica A 391: 1753-1758 (2012).