# Sznajd模型

Sketch of the 2 updating rules, social validation (top panel) and discord destroys (bottom panel), assuming that the two men in the middle have been chosen to be updated. Without loss of generality, red men (looking to the left) say no, blue men (looking to the right) say yes. The purple men can have either opinion.

The Sznajd model or United we stand, divided we fall (USDF) model is a sociophysics model introduced in 2000[1] to gain fundamental understanding about opinion dynamics. The Sznajd model implements a phenomenon called social validation and thus extends the Ising spin model. In simple words, the model states:

• Social validation: If two people share the same opinion, their neighbors will start to agree with them.
• Discord destroys: If a block of adjacent persons disagree, their neighbors start to argue with them.

The Sznajd model or United we stand, divided we fall (USDF) model is a sociophysics model introduced in 2000 to gain fundamental understanding about opinion dynamics. The Sznajd model implements a phenomenon called social validation and thus extends the Ising spin model. In simple words, the model states:

• Social validation: If two people share the same opinion, their neighbors will start to agree with them.
• Discord destroys: If a block of adjacent persons disagree, their neighbors start to argue with them.

Sznajd 模型或 United we stand，divided we fall (USDF)模型是2000年引入的社会物理学模型，旨在获得对舆论动力学的基本理解。模型实现了一种叫做社会验证的现象，从而扩展了伊辛旋转模型。简单来说，这个模型指出:

• 社会验证: 如果两个人有相同的观点，他们的邻居就会开始同意他们。不和谐破坏: 如果邻居中的一群人不同意，他们的邻居就会开始和他们争吵。

## Mathematical formulation

For simplicity, one assumes that each individual $\displaystyle{ i }$ has an opinion Si which might be Boolean ($\displaystyle{ S_i=-1 }$ for no, $\displaystyle{ S_i=1 }$ for yes) in its simplest formulation, which means that each individual either agrees or disagrees to a given question.

For simplicity, one assumes that each individual i has an opinion Si which might be Boolean (S_i=-1 for no, S_i=1 for yes) in its simplest formulation, which means that each individual either agrees or disagrees to a given question.

= = = 数学公式 = = 为了简单起见，我们假设每个个体都有一个观点 Si，这个观点 Si 在其最简单的公式中可能是布尔型的(s _ i =-1表示否，s _ i = 1表示肯) ，这意味着每个个体要么同意要么不同意给定的问题。

In the original 1D-formulation, each individual has exactly two neighbors just like beads on a bracelet. At each time step a pair of individual $\displaystyle{ S_i }$ and $\displaystyle{ S_{i+1} }$ is chosen at random to change their nearest neighbors' opinion (or: Ising spins) $\displaystyle{ S_{i-1} }$ and $\displaystyle{ S_{i+2} }$ according to two dynamical rules:

1. If $\displaystyle{ S_i=S_{i+1} }$ then $\displaystyle{ S_{i-1}=S_i }$ and $\displaystyle{ S_{i+2}=S_i }$. This models social validation, if two people share the same opinion, their neighbors will change their opinion.
2. If $\displaystyle{ S_i=-S_{i+1} }$ then $\displaystyle{ S_{i-1}=S_{i+1} }$ and $\displaystyle{ S_{i+2}=S_i }$. Intuitively: If the given pair of people disagrees, both adopt the opinion of their other neighbor.

In the original 1D-formulation, each individual has exactly two neighbors just like beads on a bracelet. At each time step a pair of individual S_i and S_{i+1} is chosen at random to change their nearest neighbors' opinion (or: Ising spins) S_{i-1} and S_{i+2} according to two dynamical rules:

1. If S_i=S_{i+1} then S_{i-1}=S_i and S_{i+2}=S_i. This models social validation, if two people share the same opinion, their neighbors will change their opinion.
2. If S_i=-S_{i+1} then S_{i-1}=S_{i+1} and S_{i+2}=S_i. Intuitively: If the given pair of people disagrees, both adopt the opinion of their other neighbor.

### Findings for the original formulations

In a closed (1 dimensional) community, two steady states are always reached, namely complete consensus (which is called ferromagnetic state in physics) or stalemate (the antiferromagnetic state). Furthermore, Monte Carlo simulations showed that these simple rules lead to complicated dynamics, in particular to a power law in the decision time distribution with an exponent of -1.5.[2]

In a closed (1 dimensional) community, two steady states are always reached, namely complete consensus (which is called ferromagnetic state in physics) or stalemate (the antiferromagnetic state). Furthermore, Monte Carlo simulations showed that these simple rules lead to complicated dynamics, in particular to a power law in the decision time distribution with an exponent of -1.5.

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =.此外，蒙特卡罗模拟表明，这些简单的规则导致复杂的动力学，特别是在决策时间分布的指数为 -1.5的幂律。

### Modifications

The final (antiferromagnetic) state of alternating all-on and all-off is unrealistic to represent the behavior of a community. It would mean that the complete population uniformly changes their opinion from one time step to the next. For this reason an alternative dynamical rule was proposed. One possibility is that two spins $\displaystyle{ S_i }$ and $\displaystyle{ S_{i+1} }$ change their nearest neighbors according to the two following rules:[3]

1. Social validation remains unchanged: If $\displaystyle{ S_i=S_{i+1} }$ then $\displaystyle{ S_{i-1}=S_{i} }$ and $\displaystyle{ S_{i+2}=S_{i} }$.
2. If $\displaystyle{ S_i=-S_{i+1} }$ then $\displaystyle{ S_{i-1}=S_{i} }$ and $\displaystyle{ S_{i+2}=S_{i+1} }$

The final (antiferromagnetic) state of alternating all-on and all-off is unrealistic to represent the behavior of a community. It would mean that the complete population uniformly changes their opinion from one time step to the next. For this reason an alternative dynamical rule was proposed. One possibility is that two spins S_i and S_{i+1} change their nearest neighbors according to the two following rules:

1. Social validation remains unchanged: If S_i=S_{i+1} then S_{i-1}=S_{i} and S_{i+2}=S_{i}.
2. If S_i=-S_{i+1} then S_{i-1}=S_{i} and S_{i+2}=S_{i+1}

= = = 修改 = = = 最终(反铁磁)状态的交替全开和全关是不现实的行为代表一个社区。这将意味着，整个人口统一地改变他们的意见，从一个时间步骤到下一个。为此，提出了另一种动力学规律。一种可能性是，两个自旋 s _ i 和 s _ { i + 1}根据以下两个规则改变它们的近邻: # 社会验证保持不变: 如果 s _ i = s _ { i + 1} ，那么 s _ { i-1} = s _ { i }和 s _ { i + 2} = s _ { i }。# 如果 s _ i =-s _ { i + 1}则 s _ { i-1} = s _ { i }和 s _ { i + 2} = s _ { i + 1}

## Relevance

In recent years, statistical physics has been accepted as modeling framework for phenomena outside the traditional physics. Fields as econophysics or sociophysics formed, and many quantitative analysts in finance are physicists. The Ising model in statistical physics has been a very important step in the history of studying collective (critical) phenomena. The Sznajd model is a simple but yet important variation of prototypical Ising system.[4]

In recent years, statistical physics has been accepted as modeling framework for phenomena outside the traditional physics. Fields as econophysics or sociophysics formed, and many quantitative analysts in finance are physicists. The Ising model in statistical physics has been a very important step in the history of studying collective (critical) phenomena. The Sznajd model is a simple but yet important variation of prototypical Ising system.

In 2007, Katarzyna Sznajd-Weron has been recognized by the Young Scientist Award for Socio- and Econophysics of the Deutsche Physikalische Gesellschaft (German Physical Society) for an outstanding original contribution using physical methods to develop a better understanding of socio-economic problems.[5]

In 2007, Katarzyna Sznajd-Weron has been recognized by the Young Scientist Award for Socio- and Econophysics of the Deutsche Physikalische Gesellschaft (German Physical Society) for an outstanding original contribution using physical methods to develop a better understanding of socio-economic problems.

2007年，Katarzyna Sznajd-Weron 因其杰出的原创性贡献，使用物理方法更好地理解社会经济问题，获得了德国物理学会德国物理学会社会经济物理学年轻科学家奖。

### Applications

The Sznajd model belongs to the class of binary-state dynamics on a networks also referred to as Boolean networks. This class of systems includes the Ising model, the voter model and the q-voter model, the Bass diffusion model, threshold models and others.[6] The Sznajd model can be applied to various fields:

• The finance interpretation considers the spin-state $\displaystyle{ S_i=1 }$ as a bullish trader placing orders, whereas a $\displaystyle{ S_i=0 }$ would correspond to a trader who is bearish and places sell orders.

The Sznajd model belongs to the class of binary-state dynamics on a networks also referred to as Boolean networks. This class of systems includes the Ising model, the voter model and the q-voter model, the Bass diffusion model, threshold models and others. The Sznajd model can be applied to various fields:

• The finance interpretation considers the spin-state S_i=1 as a bullish trader placing orders, whereas a S_i=0 would correspond to a trader who is bearish and places sell orders.

= = = 应用 = = = Sznajd 模型属于一类二元态动力学的网络，也称为布尔网络。这类系统包括 Ising 模型、选民模型和 q-voter 模型、 Bass 扩散模型、阈值模型等。Sznajd 模型可以应用于各个领域:

• 金融解释认为旋转状态的 s _ i = 1是一个看涨的交易者下单，而 s _ i = 0则对应于一个看跌的交易者下单卖出。

## References

1. Sznajd-Weron, Katarzyna; Sznjad, Jozef (2000). "Opinion evolution in closed community". International Journal of Modern Physics C. 11 (6): 1157–1165. arXiv:cond-mat/0101130. Bibcode:2000IJMPC..11.1157S. doi:10.1142/S0129183100000936. S2CID 17307753.
2. Sznajd-Weron, Katarzyna (2005). "Sznajd model and its applications". Acta Physica Polonica B. 36 (8): 2537. arXiv:physics/0503239. Bibcode:2005AcPPB..36.2537S.
3. Sanchez, Juan R. (2004). "A modified one-dimensional Sznajd model". arXiv:cond-mat/0408518.
4. Castellano, Claudio; Fortunato, Santo; Loreto, Vittorio (2009). "Statistical physics of social dynamics". Reviews of Modern Physics. 81 (2): 591–646. arXiv:0710.3256. Bibcode:2009RvMP...81..591C. doi:10.1103/RevModPhys.81.591. S2CID 118376889.
5. "Young Scientist Award for Socio- and Econophysics". Bad Honnef, Germany: Deutsche Physikalische Gesellschaft. Retrieved 15 October 2014.
6. Gleeson, James P. (2013). "Binary-State Dynamics on Complex Networks: Pair Approximation and Beyond". Physical Review X. 3 (2): 021004. arXiv:1209.2983. Bibcode:2013PhRvX...3b1004G. doi:10.1103/PhysRevX.3.021004. S2CID 54622570.