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 has an opinion Si which might be Boolean 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 and is chosen at random to change their nearest neighbors' opinion and according to two dynamical rules:
If then and. This models social validation, if two people share the same opinion, their neighbors will change their opinion.
If then and. Intuitively: If the given pair of people disagrees, both adopt the opinion of their other neighbor.
The final 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 and change their nearest neighbors according to the two following rules:
Social validation remains unchanged: If then and.
If then and
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 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 for an outstanding original contribution using physical methods to develop a better understanding of socio-economic problems.
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. The Sznajd model can be applied to various fields:
The finance interpretation considers the spin-state as a bullish trader placing orders, whereas a would correspond to a trader who is bearish and places sell orders.
