Sznajd model

• 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.

Mathematical formulation

For simplicity, one assumes that each individual ${\displaystyle i}$ has an opinion S_i 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.

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.

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.

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:

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 ${\displaystyle S_{i+2}=S_{i+1}}$