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arXiv:1006.2456v2 [physics.soc-ph] 4 Jul 2010

Consequence of reputation in the Sznajd

consensus model

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

3810-193 Aveiro

3ISB - Universidade Federal do Amazonas

69460-000Coari - AM

Brazil

Portugal

Brazil

Abstract

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-

ciophysics

1Introduction

In the last few years, the Sznajd sociophysics model [1] has been successfully

applied to many different areas, like politics, marketing and finance (for re-

views, see [2,3] and more recently [4]). The main goal of the model is the

emergence of consensus from simple microscopic rules based on an Ising-type

1nuno@if.uff.br

2fabricio forgerini@ufam.edu.br

Preprint submitted to Elsevier Science6 July 2010

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model. After the introduction of the original one-dimensional model in 2000

[1], many modifications were proposed, like the extension to the square [5],

triangular [6] and cubic lattices [7], the increase of the range of the interaction

[8] and the number of variable’s states [9,10,11], adding diffusion [9,12], and

others.

The Sznajd model defined on the square lattice was firstly studied by Stauffer

et al. [5]. 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 [13], 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 [2]. 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 [2]. 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 [2] 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 [14].

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

Section 4.

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2Model

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 [5]). 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 [15]. 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

below.

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-

changed.

(3) If all four center spins are parallel, we calculate the average reputation

of the plaquette:

average reputation =1

4

4

?

i=1

Ri,

where each term Rirepresents the reputation of one of the plaquettes’

agents

(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).

3.

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 [2], 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

i=1Ri/4.

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0

0.5

1

1.5

2

time steps (x 105)

-1

-0.5

0

0.5

1

m

d = 0.6

d = 0.4

0246

time steps (x 103)

-1

-0.5

0

0.5

1

m

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 [5]. In the

right side we show the results for d = 0.1 and d = 0.9. In these cases the system

reaches consensus.

3Numerical Results

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

m =1

N

N

?

i=1

si,(1)

where N = L2is the total number of agents and si= ±1. In the standard

Sznajd model defined on the square lattice [5], 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 [5], 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.

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104

relaxation times

101

102

# occurrences

σ = 5

σ = 10

σ = 20

101

102

L

102

103

104

105

106

107

τ

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.

0.4

0.5

0.60.7

d

0.80.91

0

0.2

0.4

0.6

0.8

1

f

L = 23

L = 31

L = 53

L = 73

L = 101

L = 121

-3-2-101

(d-dc) Lb

0

0.2

0.4

0.6

0.8

1

1.2

f La

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

[5], or some kind of special agents, like contrarians and opportunists [14], 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

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0.4

0.5

0.6 0.7

d

0.80.9 1

0

0.2

0.4

0.6

0.8

1

f

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 [16].

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 [16],

f(d,L)=L−a ˜f((d − dc) L−b) ,

dc(L)=dc+ a L−b.

(2)

(3)

The result is shown in Fig. 3 (right side), and we have found that

dc= 0.88 ± 0.01 ,(4)

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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).

4Conclusions

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 [5],

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. [5]. 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

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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

study.

Acknowledgments

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

the support.

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