Last time we have introduced you to the Sznajd model. We have also mentioned that this model was rather popular in Sociophysics around the beginning of the XXI century. This time I wanted to introduce you to one the modified Sznajd models, but I have misread the article . Well, accidents happen in science from time to time...
Nevertheless the "misread" model also looks rather nice, so I have decided to still present it. Next time I'll introduce you to the original version of this "misread" model.
Here we will introduce "intermediate" opinions into the Sznajd model. These "intermediate" opinions work in a similar manner as in the AB model. Namely, in order to transit between the extreme opinions, an agent must first transition trough intermediate opinions: he must complete the cycle \( 1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \) (or in the opposite direction). Here \( 1 \) and \( 4 \) are extremes, while \( 2 \) and \( 3 \) are intermediates. This cycle is implemented by requiring changing "social validation" rule. Note that "discord" rule is not applied in this model.
"Social validation" rule now works as follows. If two neighboring agents have the same opinion, they will convince their neighbors to copy their opinion only if neighbors opinion is within \( 1 \) of their opinion. E.g., suppose that the two random neighboring agents have opinion of \( 2 \). They will convince their neighbor to switch only if his opinion is \( 1 \) or \( 3 \). If neighbors opinion is \( 4 \), he will retain his opinion.
The original paper  suggests to leave half of the cells empty. Why? I have misread. And from what I read, I was convinced that this was some kind of strange initialization rule, which could make sense in the context of the Sznajd model. So, in my interpretation cells are left empty only during initialization: they are randomly "grown" into by agents with opinions. After this initialization we have semi-random clusters of opinion with no empty cells and later the model runs as usual.
Feel free to explore the dynamics of the "misread" four state Sznajd model. In this app you have only "one" parameter - proportion of empty cells (which are later filled in by random "growth") and agents with different opinions.
I invite you to examine the model dynamics yourself! Originally Stauffer reported that in his interpretation the "third" opinion usually won , but I am not sure if this holds true in my interpretation. Explore!
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.
- D. Stauffer. Better being third than second in search for a majority opinion. Advances in Complex Systems 5: 97-100 (2002). doi: 10.1142/S0219525902000511.