Brazilian Journal of Physics

ISSN: 0103-9733

luizno.bjp@gmail.com

Sociedade Brasileira de Física

Brasil

Atman, A. P. F.; Amin Gonçalves, Bruna

Influence of the Investor's Behavior on the Complexity of the Stock Market

Brazilian Journal of Physics, vol. 42, núm. 1-2, 2012, pp. 137-145

Sociedade Brasileira de Física

Sâo Paulo, Brasil

Available in: http://www.redalyc.org/articulo.oa?id=46423428018

How to cite

Complete issue

More information about this article

Journal's homepage in redalyc.org

Scientific Information System

Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal

Non-profit academic project, developed under the open access initiative

Braz J Phys (2012) 42:137–145DOI 10.1007/s13538-011-0057-7

STATISTICAL

Influence of the Investor’s Behavior on the Complexityof the Stock Market

A. P. F. Atman · Bruna Amin Gonçalves

Received: 12 July 2011 / Published online: 29 December 2011© Sociedade Brasileira de Física 2011

Abstract One of the pillars of the finance theory is theefficient-market hypothesis, which is used to analyzethe stock market. However, in recent years, this hy-pothesis has been questioned by a number of studiesshowing evidence of unusual behaviors in the returnsof financial assets (“anomalies”) caused by behavioralaspects of the economic agents. Therefore, it is time toinitiate a debate about the efficient-market hypothesisand the “behavioral finances.” We here introduce acellular automaton model to study the stock marketcomplexity, considering different behaviors of the eco-nomical agents. From the analysis of the stationarystandard of investment observed in the simulations andthe Hurst exponents obtained for the term series ofstock index, we draw conclusions concerning the com-plexity of the model compared to real markets. We alsoinvestigate which conditions of the investors are able toinfluence the efficient market hypothesis statements.

A. P. F. Atman (B)Departamento de Física e Matemática and InstitutoNacional de Ciência e Tecnologia—Sistemas Complexos,Centro Federal de Educação Tecnológica de Minas Gerais,CEFET-MG, Av. Amazonas 7675, Belo Horizonte,Minas Gerais, Brazile-mail: atman@dppg.cefetmg.br

A. P. F. Atman · B. A. GonçalvesMestrado em Modelagem Matemática e Computacional,Centro Federal de Educação Tecnológica de Minas Gerais,CEFET-MG, Av. Amazonas 7675, Belo Horizonte,Minas Gerais, Brazile-mail: bruna@dppg.cefetmg.br

Keywords Behavioral finance · Efficient-markethypothesis · Cellular automaton · Hurst exponent

1 Introduction

Motivated by the seminal work of H. E. Stanley andcollaborators [1], a new domain of research has de-veloped—econophysics. Using mostly statistical physicstools and the increasing calculational power of com-puters, physicists have introduced several methods tointerpret noisy and stochastic data. One of the mostpromising areas to apply this interdisciplinary knowl-edge to are stochastic time series such as the stockmarket index. Apart from finding a way to predict theindex, which was the first “gold rush” of econophysics,we focus more on trying to understand the dynamicprocess of the market and how the local behavior of theinvestors can influence the worldwide tendency of theindex.

The finance theory uses the efficient-market hypoth-esis as the ideal model system; it states that the in-vestors are rational and immediately incorporate thenew information of the market to find fair prices. Inthis way, the market becomes efficient and, because ofthis rational behavior, there would be no psychologi-cal features such as excess of confidence, indecision,and exaggeration and, therefore, a crisis would neveroccur in this market. A basis for a new theory, namedbehavioral finance, customizes concepts of psychology,sociology, and other sciences to model the investors,aiming to approximate the finance theory to the stockmarket reality. Empirical studies show that the be-havior of the investors frequently deviates from therational standards.

138 Braz J Phys (2012) 42:137–145

20 40 60 80 100

100

80

60

40

20

vert

ical

pos

ition

- i

horizontal position - j

Buy

Hold

1.500

0.5000

Sell

Fig. 1 Stationary configuration for the option of the investors;the initial condition was a random distribution of 33% for eachoption of investment and all investors imitate the neighborhoodwith P1 = 1

Compared to a complex ecosystem, the stock marketis a dynamic system in a complex network in which eachindividual (investor) searches to protect his (we will write“he” for “the investor” throughout for convenience,although we are well aware that there are many femaleinvestors as well) own safety (investment). In currenteconomic models, the agent is generally perfectly ra-tional and his target is to maximize his profit. Modelsbased on agents, as those treated by statistical physics,can go beyond the theories of the “agent” prototypeof the traditional economy because these statisticalphysics models can be used to incorporate collectivebehavior and develop a better understanding of thedynamic aspects of the financial market.

Recently, a cellular automaton model, a commontechnique in statistical physics, was proposed to sim-ulate an artificial stock market, allowing one to studyhow the market depends on the investors’ psychology[2, 3]. Yi-ming and co-authors considered two types of

0.60

0.55

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

0.60

0.55

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

0.60

0.55

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

0.60

0.55

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.001 10 100 1000 10000

1 10 100 1000 10000 1 10 100 1000 10000

1 10 100 1000 10000

perc

enta

ge o

f inv

esto

rspe

rcen

tage

of i

nves

tors

perc

enta

ge o

f inv

esto

rspe

rcen

tage

of i

nves

tors

time steps

time steps time steps

time steps

CASE(1):(0;0;1)

CASE(1):(0.3;0.3;0.4) CASE(1):(0.5;0.5;0)

CASE(1):(0.1;0.1;0.8)sellingholdingbuying

sellingholdingbuying

sellingholdingbuying

sellingholdingbuying

a b

dc

Fig. 2 Evolution of the investor’s behavior during the time for case (1). In all cases, the initial state is random with 33% for each optionof investment

Braz J Phys (2012) 42:137–145 139

behavior: the imitation, in which the investor followsthe preference of the majority of his neighbors, and theanti-imitation, in which he adopts the opposite positionof the majority of his neighbors.

We used a cellular automaton model to study themarket behavior based on the investors’ preferences.In addition to the types of investors mentioned above,we considered a third type, the indif ferent, who makeshis decisions regardless of his neighbors. To character-ize the proposed model, we used a statistical physicstechniques [1], such as the Hurst exponent, to analyzethe temporal series of the stock index produced by

the model and to verify whether the complexity of theresults is statistically coherent with real markets. Wealso compared the results with the prediction from theefficient-market hypothesis.

2 Numerical Procedure

The cellular automata were originally proposed byJ. Von Neumann and were thoroughly studied byWolfram [4]. They are discrete dynamic systems of a

CASE(1):(0;0;1) CASE(1):(0.1;0.1;0.8)

CASE(1):(0.5;0.5;0)CASE(1):(0.3;0.3;0.4)

Parameter Value

AB

R SD

-1.440

0.99981 0.00675

0.0020.4848 8 E-4

Error

Parameter Value

AB

R SD

-1.548

0.9997 0.01107

0.0030.634 0.001

Error Parameter Value

AB

R SD

-1.755

0.99745 0.04287

0.0120.846 0.006

Error

Parameter Value

AB

R SD

-1.454

0.99773 0.0243

0.0070.508 0.003

Error

w(e

)

w(e

)

scale escale e

scale e scale e

w(e

)

w(e

)1

0.1

1

0.1

1

1 10 100 1000 1 10 100 1000

1 10 100 10001 10 100 1000

0.1

10

1

0.1

0.01

120

110

100

90

802000 4000 6000 8000 10000

170

160

150

140

130

120

110

100

900 2000 4000 6000 8000 10000

160

150

140

130

120

110

100

90

80

0 2000 4000 6000 8000 10000

12011811611411211010810610410210098969492908886848280

0 2000 4000 6000 8000 10000

a b

dc

Fig. 3 Evolution of the investor behavior during the time for case (2). In all the cases, the initial state is random with 1/3 for eachoption of investment

140 Braz J Phys (2012) 42:137–145

simple construction, used as mathematics models toinvestigate problems in different areas because they arethe simplest models that are able to describe complexsystems [5]. In a general way, they consist of a uniformlattice of discrete elements (sites) that can be in k states.The condition of each site in a time step is determinedby its neighborhood at the previous step, and the tran-sition rules can be deterministic or probabilistic [6].

We represent the market as a cellular automaton intwo dimensions where each site represents an investor.The linear extension of the system is L = 100, withN = 10,000 sites (investors). We use deterministic orprobabilistic transition rules depending on the consid-ered implementation and periodic boundary conditionsin both directions. The variable St(i, j) denotes theoption of the investor at the site (i, j), at time t, fromthe following choices: Sb (to buy), Sh (to hold), and Ss

(to sell). A Moore neighborhood with eight neighborsis used, and different evolution rules are associatedto each behavior type of investors. In a general way,the state is determined by the prevailing state of theneighborhood in a specific step of time. In this way, theoption of the investor in the moment t + 1 is a functionin the majority state among its neighbors at the momentt, represented by the expression

St+1(i, j) = f(

St(i − 1, j − 1), St(i − 1, j),

St(i − 1, j + 1), St(i, j − 1),

St(i, j + 1), St(i − 1, j + 1),

St(i, j + 1), St(i + 1, j + 1)

).

(1)

0.60

0.55

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

0.60

0.55

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

0.60

0.55

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

0.60

0.55

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.001 10 100 1000 10000

1 10 100 1000 10000 1 10 100 1000 10000

1 10 100 1000 10000

perc

enta

ge o

f inv

esto

rspe

rcen

tage

of i

nves

tors

perc

enta

ge o

f inv

esto

rspe

rcen

tage

of i

nves

tors

time steps

time steps time steps

time steps

CASE(2):(0.01;0.01;0.89)

CASE(2):(0.4;0.04;0.56) CASE(2):(0.5;0.05;0.45)

CASE(2):(0.3;0.03;0.67)sellingholdingbuying

sellingholdingbuying

sellingholdingbuying

sellingholdingbuying

a b

dc

Fig. 4 Evolution of the investors behavior as a function of time for case (3). In all the cases, the initial state is random with 1/3 for eachoption of investment

Braz J Phys (2012) 42:137–145 141

The initial condition for each investor was determinedconsidering a random distribution with probability of1/3 for each option (to buy, to sell, to hold).

Three types of investor behavior were considered:“imitation,” when the choice of the investor will beidentical to the prevailing choice of its neighborhood;“anti-imitation,” when the investor will choose the op-posite of the prevailing tendency of the neighborhood(if the prevailing choice is to keep, the anti-imitationwill choose, the less prevailing of the options to buyand to sell). In the third type, named “indifferent,” thechoice is independent of the tendency of the neighborsand is determined randomly among the three possiblechoices. This third type increases the randomness of thesystem.

In simulations, the quantity of money and stocksare limited. The stock market index is altered in eachstep depending on the options of the investors, inother words, depending on the supply and demand ofstocks. The variable P1(i, j) represents the probabilitythat the agent (i, j) imitates the neighborhood, P2(i, j)represents the probability (i, j) of anti-imitation, andP3(i, j) = 1 − P1 − P2 is the probability of the agent(i, j) to be indifferent.

We considered several different combinations ofthese probabilities. First, we investigated the behaviorof the model when all investors behave identically,considering each one of the three types [7]. Interest-ingly, when all the investors chose to imitate or anti-imitate their neighbors, the system froze into random

CASE(2):(0.1;0.01;0.89)

CASE(2):(0.4;0.04;0.56)

Parameter Value

AB

R SD

-1.284

0.99573 0.03007

0.0080.458 0.004

Error

Parameter Value

AB

R SD

-1.378

0.99793 0.0364

0.0110.797 0.005

Error

w(e

)

scale e

scale e

w(e

)

1

0.1

1

1 10 100 1000

1 10 100 1000

0.1

10

120118116114112110108106104102100

98969492908886848280

20000 4000 6000 8000 10000

220

200

180

140

160

120

100

80

60

40

20

00 2000 4000 6000 8000 10000

a

c

CASE(2):(0.3;0.03;0.67)

Parameter Value

AB

R SD

-1.326

0.99854 0.02486

0.0070.649 0.003

Error

scale e

w(e

)

1

1 10 100 1000

scale e1 10 100 1000

0.1

10

130

120

110

90

100

80

70

60

0 2000 4000 6000 8000 10000

b

CASE(2):(0.5;0.05;0.45)

Parameter Value

AB

R SD

-1.552

0.997420.05064

0.0140.992 0.006

Error

w(e

)

1

0.1

100

10

130

120

110

90

100

80

70

60

0 2000 4000 6000 8000 10000

d

Fig. 5 Calculation of the Hurst exponent for the indices of thestocks in the case (1). The parameter B represents the H value.In the inset in each panel, we show the temporal evolution of the

stock index. In all cases, the initial state is random with 1/3 foreach option of investment

142 Braz J Phys (2012) 42:137–145

configurations that looked like patterns of Ising modelsclose to criticality. Figure 1 shows an example of thepattern obtained for the case in which all investorfollows the majority. The more interesting cases wereobtained when the indifferent type was included inthe system. The results of these simulations are sum-marized in the following figures and were obtainedconsidering the following choices for the probabilitiesof each type of behavior:

(1)P2 = P1; (2)P2 = 0.1P1; (3)P2 = 10P1;

3 Results and Discussion

The Hurst exponent has been introduced through thework of the biologist Harold E. Hurst in 1951, where heproposed a new statistic to distinguish a random series

from a non-random one, while examining the variousregimes of water flow in a dyke. In 1972, B. Mandelbrot[8] introduced the Hurst exponent as a tool to analyzetemporal series and calculate their fractal dimension.Edgar E. Peters used the Hurst exponent in explainingeconomic phenomena and in finance temporal series[9].

The method used most often for the calculation ofthe Hurst exponent is the detrended fluctuation analy-sis [10], which calculates the roughness around the linethat better fits a set of points. We calculated the Hurstexponent for the temporal series of the stock indexproduced by the artificial market. In the model, theindex value is denoted by X(t), and each time step isconsidered as a market-closing day.

To calculate the Hurst exponent, we divided the tem-poral series into a set of points that were not superim-posed, In, and calculated for each gap ε the best linearfit to the points inside the gap. Then, we calculated the

0.60

0.55

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0.6

0.5

0.4

0.3

0.2

0.1

0.0

1 10 100 1000 10000

perc

enta

ge o

f inv

esto

rs

time steps

1 10 100 1000 10000

time steps1 10 100 1000 10000

time steps

CASE(3):(0.02;0.2;0.78)sellingholdingbuying

sellingholdingbuying

0.60

0.55

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.001 10 100 1000 10000

perc

enta

ge o

f inv

esto

rspe

rcen

tage

of i

nves

tors

perc

enta

ge o

f inv

esto

rs

time steps

CASE(3):(0.04;0.4;0.56)

CASE(3):(0.08;0.8;0.12)CASE(3):(0.06;0.6;0.34)

sellingholdingbuying

sellingholdingbuying

a b

c d

Fig. 6 Calculation of the Hurst exponent for the indices of thestocks produced in case (2). The parameter B represents thevalue of H. In the inset of each panel, we show the graphics of

the stock index. In all cases, the initial state is random with 1/3for each option of investment

Braz J Phys (2012) 42:137–145 143

local roughness at the scale ε as the standard deviationaround the fitting line

w(ε) = 1Nε

∑In

√1T

∑t

[X(t)− (an(ε)−b n(ε)t)

]2. (2)

an and b n are the linear fit coefficients in the gap ε

and Nε is the number of gaps. The value of the Hurstexponent H is obtained from the relation

w(ε) ∼ εH , (3)

where the value of the Hurst exponent can be estimatedas the angular coefficient of the linear regression of thedata from a log–log plot. It is known that for H = 1/2,the series is decorrelated, and different values indicate

the existence of long-range correlations. When 1/2 <

H < 1, the series is positively correlated and showspersistence. For 0 < H < 1/2, the series is negativelycorrelated and shows anti-persistence. Hurst exponentswere calculated for all series of stock index for each ofthe samples.

We now consider the results from ten samples during10,000 steps of time in each case. In Figs. 2, 3, and4, we show the temporal distribution of the optionsof investment for the cases 1, 2, and 3, respectively.In Figs. 5, 6, and 7, the Hurst exponent is calculatedfrom the market index for the cases 1, 2, and 3, re-spectively, and the market index behavior is shown indetail in each of the panels. As expected, the numberof investors in each choice fluctuates randomly, and thestock index reflects this behavior, showing self-similarcharacteristics.

CASE(3):(0.02;0.2;0.78)

Parameter Value

AB

R SD

-1.607

0.99935 0.01226

0.0030.481 0.002

Error

w(e

)

scale e

1

0.1

1 10 100 1000

120118116114112110108106104102100

98969492908886848280

20000 4000 6000 8000 10000

CASE(3):(0.04;0.4;0.56)

Parameter Value

AB

R SD

-1.701

0.9984 0.01777

0.0050.464 0.002

Errorw(e

)

scale e

1

0.1

0.011 10 100 1000

120118116114112110108106104102100

98969492908886848280

20000 4000 6000 8000 10000

CASE(3):(0.06;0.6;0.34) CASE(3):(0.08;0.8;0.12)

Parameter Value

AB

R SD

-1.860

0.99748 0.02731

0.0080.542 0.004

Error Parameter Value

AB

R SD

-2.166

0.99217 0.06335

0.0180.709 0.009

Error

w(e

)

scale e

1

0.1

0.01

w(e

)

1

0.1

0.01

1 10 100 1000

scale e1 10 100 1000

120110

100

90

80

70

60

50

40

118116114112110108106104102100

98969492908886848280

20000 4000 6000 8000 10000 20000 4000 6000 8000 10000

a b

dc

Fig. 7 Calculation of the Hurst exponent of the indices from thestocks produced in case (3). The parameter B corresponds to theH estimation. In the inset of each panel, we show the graphics of

the stock index. In all cases, the initial state is random with 1/3for each option of investment

144 Braz J Phys (2012) 42:137–145

1.00

0.95

0.90hu

rst e

xpon

ent

P2

CASE(1)CASE(2)CASE(3)

CASE(1)CASE(2)CASE(3)

CASE(1)CASE(2)CASE(3)

P2

P2

0.85

0.80

0.75

0.70

0.65

0.60

0.55

0.50

0.45

0.400.0 0.1 0.2 0.3 0.4 0.5

0.020

0.018

0.016

0.014

0.012

0.010

0.008

0.006

0.004

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.0

0.5

0.4

0.3

0.2

0.1

0.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.6 0.7 0.8⏐Q

b-Q

s⏐

⏐Qb-

Qh⏐

a b

c

Fig. 8 a The average value of the Hurst exponent as a functionof P2 for each case considered. b The average difference betweenthe number of buying, Qb, and holding, Qh, positions as a

function of P2. c The average difference of the buying Qb andthe selling Qs positions

In most cases, we verified that the number of agentswho are selling and buying are generally the same, andas P2 increases, the number of agents in holding posi-tion decreases. This decreasing in the number holdingpositions directly influences the market prices, causinghuge flows. When very many people hold, the marketindex varies less.

In case 2, the effect of crowd behavior increases forthe choices to buy or to sell as P1 increases. In case 3,we observe that if the number of independent investorsis increased and the number of imitations decreases, themarket fluctuates less.

In Fig. 8, we observe the tendencies of the Hurstexponent and the option of the investors as a functionof the anti-imitation probability, P2. We observe somecorrelation between the value of the Hurst exponentand the collective crowd behavior. In case 1, clearlythe increasing of P1 (imitation investors) increases thetendency for H > 0.5, which is associated to emergingmarkets. In case 2, in which the percentage of the

imitation investors is ten times higher than that of anti-imitation investors, this crowd effect rises faster as afunction of P1 than in the other cases, and H convergesquickly to 1.

In case 3, in which the number of anti-imitations isten times higher than the number of imitations (andthe lower the number of anti-imitations, the higher thenumber of indifferent people), we observe the tendencyof H ∼ 0.5, as expected for a completely decorrelatedmarket. As P2 increases, the number of investors whohold decreases, and this alters the stability of the mar-ket, implying H > 0.5.

4 Conclusions and Perspectives

The Hurst exponent is proposed in the literature toclassify the market indices of the countries as a measureof the efficiency of the market. If long-range persis-tence is present in the return of the financial assets, the

Braz J Phys (2012) 42:137–145 145

hypothesis of the efficient market is invalid. FollowingCajueiro [11, 12], we can assume that the value ofthe Hurst exponent in developed markets—where theefficient-market hypothesis should be valid—must beclose to H = 0.5. In emerging markets, its value is lessthan this. In our artificial market, the value of theHurst exponent stays close to these expected values.In this context, the model was able to reproduce thecomplexity of the market to a degree comparable toreality.

The imitation behavior in the model can be as-sociated to the rational investor of the efficient-market hypothesis. The anti-mimic investors and theindifferent ones are the “irrational” investors of behav-ioral finance. Our results showed that the market stabil-ity is affected by the investment preference. When theinvestor behavior is affected only by other investors’behaviors, huge market fluctuactions are expected—acrowd effect. When there are many imitation investors,the Hurst exponent exceeds 0.5; the efficient-markethypothesis is then invalid. The Hurst number is closestto 0.5 the higher the number of indifferent investorsand the lower the number of imitation investors, asexpected for the developed markets (efficient markets).Thus, the stock market stability is the highest when theinvestor’s behavior is most independent.

Acknowledgements The authors would like to thank the stim-ulating discussions with the professors Rodrigo Nogueira Car-doso, Felipe Paiva, and Renato dos Santos. The authors thankFAPEMIG, CAPES, and CNPq for the financial support. APFAalso thanks for grant Propesq/CEFET-MG 076.

References

1. R.N. Mantegna, H.E. Stanley, An Introduction to Econo-physics: Correlations and Complexity in Finance. (CambridgeUniversity Press, Cambridge, EU 1999)

2. W. Yi-ming, Y. Shang-jun, F. Ying, W. e Bing-Hong, Thecellular automaton model of investment behavior in the stockmarket. Physica A 325, 507–516 (2003)

3. W. Yi-ming, Y. Shang-jun, F. Ying, W. e Bing-Hong, Theeffect of investor psychology on the complexity of stock mar-ket: An analysis based on cellular automaton model. Comput.Ind. Eng. 56, 63–69 (2009)

4. S. Wolfram, Statistical mechanics of cellular automata. Rev.Mod. Phys. 55(3), 601–644 (1983)

5. A.P.F. Atman, Aspectos Fractais em Sistemas Complexos.Ph.D. Thesis, UFMG, Belo Horizonte (2002)

6. N. Boccara, Modeling Complex Systems, 1◦ edn. (Springer,Chicago, EU 2004)

7. B.A. Gonçalves, A.P.F. Atman, Comportamento do investi-dor na complexidade do mercado de ações. in Proceed-ings do XIII Encontro de Modelagem Computacional. (NovaFriburgo, v. único, 2010)

8. A. Carbone, G. Castelli, H.E. Stanley, Time-dependent Hurstexponent in financial time series. Physica A 344, 267–271(2004)

9. R.L. Costa, G.L. Vasconcelos, Long-range correlations andnonstationarity in the Brazilian stock market. Physica A 336,231–248 (2003)

10. J.G. Moreira, J.K.L. Silva, S.O. e Kamphorst, On the Fractaldimension of self-Affine profiles. J. Phys. A Math. Gen. 27,8079–8089 (1994)

11. D.O. Cajueiro, B.M. Tabak, Testing for long-range depen-dence in world stock markets. Chaos, Solitons and FractalsN. 37, 918–927 (2008)

12. D.O. Cajueiro, B.M. Tabak, The Hurst exponent over time:testing the assertion that emerging markets are becomingmore efficient. Physica A. N. 336, 521–537 (2004)