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.
Mathematical formulation
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For simplicity, one assumes that each individual
has
an opinion S
i
which might be
Boolean
(
for
no
,
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
and
is chosen at random to change their nearest neighbors' opinion (or:
Ising spins
)
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.
Findings for the original formulations
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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]
Modifications
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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
and
change their nearest neighbors according to the two following rules:
[3]
- Social validation
remains unchanged: If
then
and
.
- If
then
and
Relevance
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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 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]
Applications
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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
as a bullish trader placing orders, whereas a
would correspond to a trader who is bearish and places sell orders.
References
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External links
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