Sznajd (United we stand, divided we fall) model

Sznajd model, also known as united we stand, divided we fall model, is another classical model in opinion dynamics. It was proposed in 2000 by two Polish scientists in [1] and since then it was heavily studied both by various groups of sociophysicists. It was used not only in the usual generic scenarios (e.g., exploring how fast the opinions converge), but also helped to predict Polish elections of 2015 [2].

The model

Sznajd model itself is rather simple. In its original formulation [1] it uses only two rules: social validation rule and discord rule.

Unlike in the most other models of opinion dynamics these rules apply not only to two neighboring agents, but to their immediate neighbors as well. If the two neighboring agents agree (share the same opinion), their neighbors will also start to agree. If the two neighboring agents disagree (have differing opinions), their neighbors will also disagree with the two agents.

The first rule here is called social validation rule, because the opinion of the agreeing agents gets validated. This rule is a lot like ferromagnetic interaction in the Ising model and also similar to the copying (herding) interactions in the voter model. Actually it was shown that this model is equivalent to voter model with one caveat - the opinion spreads not through the immediate neighbors (\( i \) influencing \( i+1 \)), but through the second neighbors (\( i \) influencing \( i+2 \)). This rule also stands for the "united we stand" part in the alternative naming of the model.

The second rule in this model (not used in the later iterations of the model) is called discord rule, because disagreement (discord) spreads. This rule is a lot like anti-ferromagnetic interaction in the Ising model. Using this rule creates chessboard patterns. This rule stands for the "divided we fall" part in the alternative naming of the model.

Similarity to the Ising and the voter models suggests what can be expected from the model. This model will reach a stationary state with average opinion, \( M \), of \( -1 \), \( 0 \) or \( 1 \).

HTML 5 app

The app below allows you to explore the dynamics of the Sznajd model. You can enable and disable usage of the both of the discussed rules as well as select the probability for an agent to have a positive opinion at \( t=0 \).

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

• K. Sznajd-Weron, J. Sznajd. Opinion evolution in closed community. International Journal of Modern Physics C 11: 1157-1165 (2000). doi: 10.1142/S0129183100000936. arXiv:cond-mat/0101130 [cond-mat.stat-mech].
• P. Sobkowicz. Quantiative agent based model of opinion dynamics: Polish Elections of 2015. PLoS ONE 11(5): e0155098 (2016). doi: 10.1371/journal.pone.0155098.