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Simulation of majority rule disturbed by power-law noise on directed and undirected Barabas

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0802.1476v1 [cond-mat.dis-nn] 11 Feb 2008Simulationofmajorityruledisturbedbypower-lawnoiseon

directedandundirectedBarab´asi-Albertnetworks

F.W.S.Lima

DepartamentodeF´ısica,UniversidadeFederaldoPiau´ı,64049-550,Teresina

-PI,Brazil

e-mail:wel@ufpi.br

Abstract:OndirectedandundirectedBarab´asi-AlbertnetworkstheIsingmodel

withspinS=1/2inthepresenceofakindofnoiseisnowstudiedthroughMonte

Carlosimulations.ThenoisespectrumP(n)followsapowerlaw,whereP(n)is

theprobabilityofflippingrandomlyselectnspinsateachtimestep.Thenoise

spectrumP(n)isintroducedtomimictheself-organizedcriticalityasamodel

influenceofacomplexenvironment.Inthismodel,differentfromthesquarelattice,

theorder-disorderphasetransitionoftheorderparameternotisobserved.For

directedBarab´asi-Albertnetworksthemagnetisationtendstozeroexponentially

andundirectedBarab´asi-Albertnetworks,itsremainconstant.Keywords:MonteCarlosimulation,spins,networks,Ising.

Introduction

ThispaperdealswithIsingspinondirectedandundirectedBarab´asi-

Albert(BA)networksinthepresenceofanoise.SumourandShabat[1,2]

investigatedIsingmodelswithspinS=1/2ondirectedBAnetworks[3]with

theusualGlauberdynamics.Nospontaneousmagnetisationwasfound,in

contrasttothecaseofundirectedBAnetworks[4,5,6]whereasponta-

neousmagnetisationwasfoundbelowacriticaltemperaturewhichincreases

logarithmicallywithsystemsize.LimaandStauffer[8]simulateddirected

square,cubicandhypercubiclatticesintwotofivedimensionswithheat

bathdynamicsinordertoseparatethenetworkeffectsofdirectedness.They

alsocompareddifferentspinflipalgorithms,includingclusterflips[9],for

Ising-BAnetworks.Theyfoundafreezing-inofthemagnetisationsimilarto

[1,2],followinganArrheniuslawatleastinlowdimensions.Thislackof

aspontaneousmagnetisation(intheusualsense)isconsistentwiththefact

thatifonadirectedlatticeaspinSjinfluencesspinSi,thenspinSiinturn

doesnotinfluenceSj,andtheremaybenowell-definedtotalenergy.Thus,

theyshowthatforthesamescale-freenetworks,differentalgorithmsgive

1differentresults.RecentlyStaufferandKu󰁍lakowski[10]simulatedtheIsing

two-dimensionalferromagnetinthepresenceofaspecialkindofnoisedif-

ferentfromthetraditionaltemperatureandBoltzmannprobabilities,where

thenoiseconsistofateachiterationofaL×Lsquarelatticewithfour

neighboursforeachsites,besidestheabovemajorityruletheyselectntimes

randomlyaspinandflipit.TheprobabilitydistributionfunctionP(n)of

thesenumbersnistakenasapowerlaw,

P(n)∝1/nα.(1)

02004006008001000time0.0107.0107.3107.5107.6107.7

magnetisationRelaxation, 500000 sites, 100 samples summed

Figure1:Summedmagnetisationsversustime,T=1,α=1,andm=2for

directed(BA)networks.

Inordertogetthisdistribution,theydeterminedrandomnumbersr,

homogeneouslydistributedbetweenzeroandone,andthentook

n=TL2r1/(1−α),(2)

2

02004006008001000time−107.30.0107.3107.6107.8

magnetisations100 samples, 500000,T: 0.001, 0.01, 0.05, 0.1, 0.5, 1.0 from top

02004006008001000time−107.30.0107.3107.6107.8

magnetization100 samples 500000, T: 0.1, 0.2, 0.3, 0.5, 1. from top

Figure2:Summedmagnetisationsversustimefor0.01≤T≤1.0,α=1

andm=2(left)and0.1≤T≤1.0,α=1andm=7(right)fordirected

(BA)networks.

forα<1,where1≤n≤L2and

n=Texp(rln(L2))(3)

forα=1andTdeterminestheamplitudeofthenoise.Herewesimu-

latedthesamemodeldescribedaboveondirectedandundirected(BA)net-

worksandourresultsaredifferentfromtheresultsobtainedbyStaufferand

Ku󰁍lakowski[10].

Results

InFig.1weshowthedependenceofthesummedmagnetisationM=󰀁iSiversustimeforα=1,T=0.01(eq.3)andconnectionnumberm=2

ondirected(BA)networkswith500000sites,whenwestartedwithallspins

up.Mrelaxesexponentiallytowardszero.InFig.2weplotsamepicture

ofFig.1,butforseveralvaluesofTandconnectionnumbersm=2(left)

andm=7(right),whereforsmallerTthemagnetisationtriestoremain

positive.IntheFig.3weplotthedependenceofthesummedmagnetisation

versustimeforα=1,severalvaluesofT(left),andforT=10(right)even

upto10000MonteCarlostepswithm=2onundirected(BA)networks

with10000sites.Thesummedmagnetisationremainspositiveanddoes

notdecaytowardszero.Itkeepsfluctuatingaroundapositivevalueofthe

magnetisationdependingonthevalueofT.

3

02004006008001000time0.00105.00105.30105.48105.60105.70105.78105.85105.90105.95

magnetisation100 samples 10000, T: 1, 10, 100, 1000 from top

0200040006000800010000time0.00105.00105.30105.48105.60105.70105.78105.85105.90105.95106.00

magnetisation100 samples 10000, T=10, m=2

Figure3:Summedmagnetisationsversustimefor1≤T≤1000.,α=1

andm=2(left)andforT=10,α=1andm=2(right)forundirected

(BA)networks.

InFig.4themagnetisationisshownasafunctionoftimeforonesample

forT=1,α=1andm=2ondirected(left)andundirected(right)(BA)

networks.Fordirected(BA)networksthesummedmagnetisationchanges

betweenpositiveandnegativevalueswithtime,andforundirected(BA)net-

worksthemagnetisationfluctuatesaroundapositivevalue.InFig.5we

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On a five to three vote, the Supreme Court knocked out much of Arizona’s

immigration law Monday—a modest policy victory for the Obama Administration. But

on the more important matter of the

Constitution, the decision was an

8-0 defeat for the Administration’s effort to upset the balance of power

between the federal government and the states.

单词&词组搭配

• constitution [ˌkɑːnstɪˈtuːʃn] （出现4次）n.宪法；[僻义] n.身体素质

• defeat [dɪˈfiːt] （出现2次）n.失败

• knock out（破坏，使...失效）

• upset the balance of power between（打破两者之间力量的平衡）

本段翻译

最高法院周一以5:3的投票结果裁定亚利桑那州移民法多项条款无效——对奥巴马政府来说是场小政策胜利。不过，在宪法这一更重要的问题上，最高法院以8:0的一致表决使企图打破联邦政府与各州之间权力平衡的奥巴马政府完败。

In Arizona v. United States, the majority overturned three of the

four contested provisions of Arizona’s controversial plan to have state and

local police enforce federal immigration law. The Constitutional principles that

英语一阅读刷题2013年text4课堂笔记

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1段 On a five to three vote, the Supreme Courtknocked out（否定；推翻）much of Arizona’s immigration law Monday—a modestpolicy victory for the Obama Administration.

Butonthe more important matter of the Constitution, the decision was an 8-0 defeat for the

Administration’s effort（企图；坏事）to upset the balance of power between the federal

government and the states.

2段 In Arizona v. United States, the majority overturned three of the fourcontested

provisions of Arizona’s controversial plan to have state and local police enforce federal

immigration law. The Constitutional principles （thatWashington alone has the power to

“establish a uniform Rule of Naturalization”）and thatfederal laws precede state laws are

noncontroversial. Arizona had attempted to fashion state policies that ran parallel to the

existing federal ones.

上海市嘉定区封浜高级中学2024-2025学年高二上学期期中考试英语试卷

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一、语法填空

Directions: After reading the passage below, fill in the blanks to make the passage coherent and

grammatically correct. For the blanks with a given word, fill in each blank with the proper form of

the given word; for the other blanks, use one word that best fits each blank.

Are you on 1 diet? Then maybe you should consider 2 (sit) by the window

the next time you eat out. A new book claims that 3 you sit in a restaurant can affect how

much you consume.

A new research found that people sitting farthest from the front door ate the fewest salads

and were 73 percent 4 (likely) to order dessert.

The diners 5 (seat) at a dark table ate heavier food and ordered more of it, 6

those sitting at bright bar tables ordered more salads and fewer desserts.