arXiv:1006.2456v2 [physics.soc-ph] 4 Jul 2010
Consequence of reputation in the Sznajd
Nuno Crokidakis1,2and Fabricio L. Forgerini2,3
1Instituto de F´ ısica - Universidade Federal Fluminense
Av. Litorˆ anea s/n
24210-340 Niter´ oi - RJ
2Departamento de F´ ısica, I3N - Universidade de Aveiro
3ISB - Universidade Federal do Amazonas
69460-000Coari - AM
In this work we study a modified version of the Sznajd sociophysics model. In partic-
ular we introduce reputation, a mechanism that limits the capacity of persuasion of
the agents. The reputation is introduced as a score which is time-dependent, and its
introduction avoid dictatorship (all spins parallel) for a wide range of parameters.
The relaxation time follows a log-normal-like distribution. In addition, we show that
the usual phase transition also occurs, as in the standard model, and it depends on
the initial concentration of individuals following an opinion, occurring at a initial
density of up spins greater than 1/2. The transition point is determined by means
of a finite-size scaling analysis.
Keywords: Population dynamics, Phase Transition, Computer Simulation, So-
In the last few years, the Sznajd sociophysics model  has been successfully
applied to many different areas, like politics, marketing and finance (for re-
views, see [2,3] and more recently ). The main goal of the model is the
emergence of consensus from simple microscopic rules based on an Ising-type
Preprint submitted to Elsevier Science6 July 2010
model. After the introduction of the original one-dimensional model in 2000
, many modifications were proposed, like the extension to the square ,
triangular  and cubic lattices , the increase of the range of the interaction
 and the number of variable’s states [9,10,11], adding diffusion [9,12], and
The Sznajd model defined on the square lattice was firstly studied by Stauffer
et al. . Considering that a 2 × 2 plaquette with all spins parallel can con-
vince their eight neighbors (we can call this Stauffer’s rule), the authors found
a phase transition for initial density of up spins d = 1/2. The two-dimensional
model was also considered in a randomly diluted square lattice , where
every site may carries one spin or is empty. A situation with long-range cor-
relations between the occupation of various lattice sizes was also considered,
but in both cases the results are very similar to those on regular lattices . A
more realistic situation is to consider a probability of persuasion. The Sznajd
model is robust against this situation: if one convinces the neighbors only with
some probability p, and leaves them unchanged with probability 1 − p, still a
consensus is reached after a long time . Models that consider many different
opinions (using Potts’ spins, for example) or defined on small-world networks
were studied in order to represent better approximations of real communities’
behavior (see  and references therein). In order to avoid full consensus in
the system, and makes the model more realistic, Schneider introduced oppor-
tunists and persons in opposition .
Unfortunately the dynamics of social relationships in the real world shows
a large number of details that are commonly neglected in some theoretical
models. In order to introduce a more realistic feature, we have considered
in this work a reputation mechanism that limits the capacity of persuasion
of the agents. It is expected that the inclusion of reputation in the Sznajd
model turns it closer to a real social system, where not only the number of
individuals with same opinion matters. We believe that the reputation of the
agents who holds same opinion is an important factor in persuasion across the
community. In other words, it is realistic to believe that individuals change
their opinions if they are influenced by high-reputation persons. In fact, we
show that a democracy-like situation, with a ferromagnetic ordering with not
all spins parallel, is possible considering simple microscopic rules.
This Letter is organized as follows. In Section 2 we present the model and
define their microscopic rules. The numerical results as well as the finite-size
scaling analysis are discussed in Section 3. Our conclusions are presented in
We have considered the Sznajd model defined on a square lattice with linear
size L (i.e., L2agents). Our model is based on the Stauffer’s rule (rule Ia of
the model in ). Moreover, an integer number (R) labels each player and
represents its reputation across the community, in analogy to the Naming
game model considered by Brigatti . The reputation is introduced as a
score which is time-dependent. The agents start with a random distribution
of the R values, and during the time evolution, the reputation of each agent
changes according to its capacity to be persuasive, following the rules explained
At each time step, the following microscopic rules control our model:
(1) We randomly choose a 2 × 2 plaquette of four neighbors.
(2) If not all four center spins are parallel, leaves its eight neighbors un-
(3) If all four center spins are parallel, we calculate the average reputation
of the plaquette:
average reputation =1
where each term Rirepresents the reputation of one of the plaquettes’
(4) In this case, we compare the reputations of each one of the eight neighbors
with the average reputation. If the reputation of a neighbor is less than
the average one, this neighbor follows the plaquette orientation. For each
persuaded individual, each one of the plaquette agents’ increase their
reputations by 1 (in other words, the average reputation of the plaquette
is increased by 1).
Thus, even in the case where all plaquettes’ spins are parallel, a different
number of agents may be convinced, namely, 8, 7, 6, ..., 1 or 0 ones. As pointed
by Stauffer , we can imagine that each agent in the Sznajd model carries an
opinion, that can either be up (e.g. Republican) or down (e.g. Democrat) and
that represents one of two possible opinions on any question. The objective of
the agents in this game is to convince their neighbors of their opinion. One
can expect that, if a certain group of agents convince many other agents, their
persuasion abilities increase. Thus, the inclusion of reputation in the model
may capture this real-world characteristic.
3Notice that we will consider the integer part of the ratio?4
time steps (x 105)
d = 0.6
d = 0.4
time steps (x 103)
d = 0.9
d = 0.1
Fig. 1. (Colour online) Time evolution of the magnetization for L = 53, initial
densities of up spins d = 0.4 and d = 0.6 and different samples (left side). We can
see that the steady states show situations where the total consensus is not obtained,
in opposition of the standard Sznajd model defined on the square lattice . In the
right side we show the results for d = 0.1 and d = 0.9. In these cases the system
In the simulations, the initial values of the agents’ reputation follow a gaussian
distribution centered at 0 with standard deviation σ = 5
previous works on the Sznajd model, we can start studying the time evolution
of the magnetization,
4. Following the
where N = L2is the total number of agents and si= ±1. In the standard
Sznajd model defined on the square lattice , the application of the Stauffer’s
rule, where a 2×2 plaquette convince its eight neighbors if all spins are parallel,
with initial density of up spins d = 1/2 leads the system to the fixed points
with all up or all down spins with equal probability. For d < 1/2 (> 1/2) the
system goes to a ferromagnetic state with all spins down (up) in all samples,
which characterizes a phase transition at d = 1/2 in the limit of large L.
As pointed by the authors in , fixed points with all spins parallel describe
the published opinion in a dictatorship, which is not a commom situation
nowadays. However, ferromagnetism with not all spins parallel corresponds
to a democracy, which is very commom in our world. We show in Fig. 1 the
behavior of the magnetization as a function of time in our model, considering
the above-mentioned rules. In the left side of Fig. 1, we show a value of d > 1/2
(< 1/2), and one can see that the total consensus with all spins up (down) will
not be achieved in any sample, indicating that (i) a democracy-like situation
4In the following, we discuss the effects of increasing the standard deviation of the
initial distribution of agents’ reputation.
σ = 5
σ = 10
σ = 20
σ = 5
σ = 10
σ = 20
Fig. 2. (Colour online) Left side: log-log plot of the histogram of relaxation times
for L = 53 and d = 0.8, obtained from 104samples, with agents’ initial reputations
following a gaussian distribution with different standard deviations σ. The distri-
bution is compatible with a log-normal one for all values of σ, which corresponds
to the observed parabola in the log-log plot. Right side: average relaxation time τ,
over 104samples, versus latice size L in the log-log scale. The straight line has slope
5/2. The result is robust with respect to the choice of different σ values.
L = 23
L = 31
L = 53
L = 73
L = 101
L = 121
L = 23
L = 31
L = 53
L = 73
L = 101
L = 121
Fig. 3. (Colour online) Left side: fraction f of samples which show all spins up when
the initial density d is varied in the range 0.4 < d < 1.0, for some lattice sizes L.
The total number of samples is 1000 (for L = 23, 31 and 53), 500 (for L = 73)
and 200 (for L = 101 and 121). Right side: the corresponding scaling plot of f. The
best collapse of data was obtained for a = 0.035 ± 0.002, b = 0.444 ± 0.002 and
dc= 0.88 ± 0.01.
is possible in the model without the consideration of a mixing of different rules
, or some kind of special agents, like contrarians and opportunists , and
(ii) if a phase transition also occurs in our case, the transition point will be
located somewhere at d > 1/2. Also in Fig. 1 (right side), we show situations
where the consensus is obtained with all spins up (for d = 0.9) and with all
spins down (for d = 0.1).
As in the previous studies of the Sznajd model [1,2,5,16], the relaxation time,
i.e., the time needed to find all the agents at the end having the same opinion,
depends on the model’s parameters. The distribution of the number of sweeps
through the lattice, averaged over 104samples, needed to reach the fixed point
σ = 5
σ = 20
σ = 100
Fig. 4. (Colour online) Fraction f of samples which show all spins up when the
initial density d is varied in the range 0.35 < d < 1.0, for L = 23, 1000 samples and
some different values of σ. This result show that the increase of σ do not change
the behavior of f.
is shown in Fig. 2 (left side). We can see that the relaxation time distribution
is compatible with a log-normal one for all values of the standard deviation σ,
which corresponds to a parabola in the log-log plot of Fig. 2 (left side). The
same behavior was observed in other studies of the Sznajd model [5,14,16]. In
the right side of Fig. 2 we show the average relaxation time τ (also over 104
samples) versus latice size L in the log-log scale. We can verify a power-law
relation between these quantities in the form τ ∼ L5/2, for all values of the
standard deviation, which indicates that the result is robust with respect to
the choice of different σ values. Power-law relations between τ and L were also
found in a previous work on the model .
In order to analyze in more details the phase transition, we have simulated
the system for different lattice sizes L and we have measured the fraction of
samples which show all spins up when the initial density of up spins d is varied
in the range 0.4 < d < 1.0. We have considered 1000 samples for L = 23, 31
and 53, 500 samples for L = 73 and 200 samples for L = 101 and 121. The
results are shown in Fig. 3 (left side). One can see that the transition point is
located somewhere in the region d > 1/2, as above discussed. In order to locate
the critical point, we performed a finite-size scaling (FSS) analysis, based on
the standard FSS equations ,
f(d,L)=L−a ˜f((d − dc) L−b) ,
dc(L)=dc+ a L−b.
The result is shown in Fig. 3 (right side), and we have found that
dc= 0.88 ± 0.01 ,(4)
in the limit of large L. The critical point occurs at d > 1/2, different of the
Sznajd model without reputation defined on the square lattice. This fact may
be easily understood: at each time step, the randomly chosen 2×2 plaquette
may convince 8, 7, 6, ..., 1 or 0 neighbors, even if the plaquettes’ spins are
parallel. In the standard model, if the plaquettes spins’ orientations are the
same, 8 neighbors are convinced immediately, thus it is necessary a smaller
initial density of up spins to the system reaches the fixed point with all spins
up. However, the existence of a critical value dc< 1.0 indicates that a phase
transition also occurs in our version of the Sznajd model. This result is robust
with respect to the choice of different values of σ (see Fig. 4).
In this work, we have analyzed the effects of the introduction of agents’ rep-
utation in the square lattice Sznajd model. The reputation is introduced as
a score (R) which is time-dependent. The agents start with a gaussian distri-
bution of the R values, and during the time evolution, the reputation of each
agent changes according to its capacity of persuasion, following the model’s
rules. We expect that the consideration of agents’ reputation makes the Sz-
najd model more realistic. In fact, take into account simple microscopic rules,
a democracy-like situation emerges spontaneously in the model, for a wide
range of the initial densities d of up spins. The consensus with all spins up
(down) is obtained only for large (small) values of d.
We performed Monte Carlo simulations on square lattices with linear sizes up
to L = 121 and typically 103− 104samples. As in the standard model ,
we found a log-normal distribution of the relaxation times. In addition, the
average relaxation time τ depends on the lattice size in a power-law form,
τ ∼ L5/2, which is independent of the standard deviation σ of the gaussian
distribution of the agents’ initial reputation. The system also undergoes a
phase transition, as in the traditional model, but the critical initial density of
up spins was found to be dc= 0.88, in the limit of large lattices, greater than
1/2, the value found by Stauffer et al. . This fact may be easily understood:
at each time step, the randomly chosen 2×2 plaquette may convince 8, 7, 6, ...,
1 or 0 neighbors, even if the plaquettes’ spins are parallel. In the standard case,
if the plaquettes spins’ orientations are the same, 8 neighbors are convinced
immediately, thus it is necessary a smaller initial density of up spins to the
system reaches the fixed point with all spins up. The simulations indicate that
the observed phase transition is robust with respect to the choice of different
values of σ.
As extensions of this work, it could be interesting to explore the role of the
system topology. Different complex topologies could be studied for agents
embedded on more realistic networks, like small-world ones. Agents with more
than two opinions, using Potts’ spins, for example, can also be interesting to
The authors are grateful to Edgardo Brigatti and Adriano O. Sousa for dis-
cussions about social models. N. Crokidakis would like to thank the Brazilian
funding agency CAPES for the financial support at Universidade de Aveiro at
Portugal. Financial support from CNPq is also acknowledge. F. L. Forgerini
would would like to thank the ISB - Universidade Federal do Amazonas for
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