Eur.Phys.J.B25,123–127(2002)
DOI:10.1140/e10051-002-0013-y
Election results and the Sznajd model on Barabasi network
乖张什么意思A.T.Bernardes1,2,D.Stauffer2,a,and J.Kert´e sz3
1Departamento de F´ısica,Universidade Federal de Ouro Preto,Campus do Morro do Cruzeiro,35400-000,
Ouro Preto-MG,Brazil
2Institute for Theoretical Physics,Cologne University,50923K¨o ln,Germany
3Department of Theoretical Physics,Budapest University of Technology and Economics,Budafoki´u t8,
1111,Budapest,Hungary
Received19September2001and Received infinal form2November2001
轮船图片大全图片Abstract.The network of Barabasi and Albert,a preferential growth model where a new node is linked to the old ones with a probability proportional to their connectivity,is applied to Brazilian election results.
The application of the Sznajd rule,that only agreeing pairs of people can convince their neighbours,gives
a vote distribution in good agreement with reality
PACS.05.50.+q Lattice theory and statistics(Ising,Potts,etc.)–89.65.-s Social systems–
02.50.-r Probability theory,stochastic process,and statistics
1Introduction
Nowadays,it has been a matter of increasing interest[1]to
apply the fundamentals of the theories of complex systems
in many different disciplines,not only in physical sciences,
but even in social sciences,from economy to education[2]
or sociology.The main point is that social systems,like
natural ones,are constituted of great number of individ-
uals,which–generally–have local interactions between
them.Sometimes,social networks behaviour can be deter-
mined also by the action of external actors,which might
be mimicked by externalfields in our model.
Elections are process where many individuals inter-
act between them.It is a dynamical convincing process,
where we have at the same time the interaction between
neighbours and external influence(political advertis-
ing,campaigns etc).In Brazil,in proportional elections
(deputies or city councillors)the voters vote directly for
the candidates and not for the parties.They can vote for
a party,but it is not frequent.Some elections occur with
a large number of voters:In some states or in the largest
cities one has a number of voters in the order of magni-
tudes of millions or tens of millions.So,the elections
qq修改密码>凌讯are a social phenomenon which prents the basic char-
acteristics of complex systems.One of the features is
that they are scale-free phenomena.This feature has been
obrved by Costa-Filho et al.[3],who showed that the
刹车油多久更换
distribution of the number of votes obtained by different
candidates for the1998elections in Brazil follow a power
124The European Physical Journal B
look at intermediate times when there are still many dif-ferent domains or correlated sites in the system.
In the Sznajd model,small ts of people influence the opinions of their nearest neighbours if and only if all people within the original t agree.On a chain,this t is a bond with two people at its ends[7,9].On the square lattice with[10]or without[8]disorder,it can be such a bond or a plaquette of four neighbouring spins(people). This plaquette rule is called rule Ia in[8]and the bond rule,ud in the prent paper,is called rule IIa in[8]. Thus if all four plaquette members or two bond members share the same opinion,they convince their neighbours of this opinion;otherwi the neighbours remain unchanged.
对宝宝的寄语However,as shown by Barabasi and Albert re-cently[14],social relations must be reprented by net-works instead of lattices.Networks of interactions(www, author’s collaboration in scientific papers,act
or’s collabo-rations infilms)show the common feature of scale-free be-haviour.In order to reprent this main feature,Barabasi and Albert introduced a model for evolving networks.Nu-merous papers ud this model for a variety of purpos, e.g.[15].Starting with few nodes(which may reprent actors,authors,web sites)connected to each other,more and more nodes are added to the network,each node con-necting to an already connected one,with the probability to connect to a node being proportional to the number of previous nodes which are already connected to it.
In Brazil the voting process is much more bad on the relation between candidate/voter than on the parties. Thus,our idealized version of the voting process can omit the role of the parties.Another aspect is that it was clear for us that the candidates do not start with the same social weight.This determines the result of the elections,since candidates with more social visibility or better conditions to campaign are more likely to be elected.So,we have to introduce some differentiation between the candidates in the beginning of the simulation.
刘梓晨微博
In this paper,we performed simulations on a three-dimensional version of the previous one-dimensional[7] and two-dimensional[8]Sznajd model.We combine it with a network model for elections bad on the model of Bernardes et al.[5].Unlike thisfirst version[5]and its three-dimensional variant where a probability to con-vince had to be introduced,equation(1)below,in order t
o produce some differentiation between the candidates,with the Barabasi network the same result is obtained from the combination of the different number of neighbours of the nodes without this probability.
In the next ction we prent the models we have ud,followed by the results.Both of the models u networks connecting the voters;one network is a simple cubic lattice,the other a Barabasi network.After that,we conclude.
2Models and results
We have simulated two models in the prent work.The first one is a3d version of that simulated previously[5]. The cond is a Barabasi network version.2.1Simple cubic lattice
In this work,we ud a modified version of the Sznajd model(rule IIa in[8]):A pair of neighbours in agreement convinces its ten nearest neighbours to the same opinion.
A cubic lattice of size L×L×L reprents the t of voters.A number N tot of candidates,N tot L3, isfixed in the beginning of the simulation.The value n=1,2,...,N tot of a site S on the lattice will reprent that this voter prefers that candidate n.The model has two different stages:First,we produc
e the initial condi-tion and,after that,we perform the simulation of the electoral campaign(only voters can influence other voters, a la Sznajd).As in real elections,we do not wait for a kind of equilibrium state,but count the votes at some intermediate time.Basically what we are doing is the analysis during the transient time.As in real elections,the candidates have different initial chances of being voted for (reprenting more money for electoral campaigns,more initial visibility etc.).This is modelled by a probability P c of convincing,calculated from the label n of the candidate
P c=(n/N tot)2.(1) It means that the higher is the label n of a candidate,the higher is the probability of convincing a voter.
In thefirst stage,we started with all the sites with value zero,meaning that there are no committed voters. Then,we visit all the sites exactly once,in random or-der.For each visit,we try to convince the voter to adopt a candidate,chon at random.A random number r is generated and compared with P c.If r≤P c the candidate is accepted by that voter.If the candidate convinces the voter,this voter tries to convince the neighbouring sites. Once again,we throw the dice and compare a new random number with P c.If successful,r≤P c,the voter will try to convince the neighbourhood as follows:We check all the six neighbouring sites;for each that has the same value of the candidate chon before,all the ten neighbouring sites of this bond of two sites will assume the same value(as in the us
ual Sznajd prescription).If nobody has chon the same candidate,only the originally lected voter is committed to this candidate.
In the cond stage,a usual Sznajd process is per-formed without using the complication from the probabil-ity P c.(We thus assume all voters to be equal and restrict the probability P c to describe the convincing power of the candidates only.)We go to random sites on the lattice.
A neighbouring site is chon at random and we check if the two sites have the same value(they prefer the same candidate).In that ca,all the ten neighbours change to vote in that candidate.
Figure1shows,just as in real life[3],deviations from a simple power law for both very large numbers of votes and very small numbers.In between,however,the simulations are compatible with the hyperbolic law
N(v)∝1/v(2)
A.T.Bernardes et al.:Election results and the Sznajd model on Barabasi network
125
1e-08
1e-071e-061e-050.00010.0010.010.1110101001000100001000001e+061e+07
n u m b e r o f c a n d i d a t e s
votes
Vote distribution, 10 simple cubic lattices 301^3, 200 candidates at t=50(x) and 100 (+); slope -1Fig.1.Distribution N (v )of the number of candidates getting v votes each on the simple cubic lattice,after 50and 100iterations.Election of 200candidates by 27million
voters.
Fig.2.Distribution N (v )for half a million nodes on the Barabasi network,where each previously added node bonds to five previously added nodes.Election of 1000candidates (+).The number of votes in real elections (×:state of Minas Gerais in Brazil 1998)is multiplied by ten for better comparison.
obrved in reality for the number N of candidates having v votes each.(Here and in Fig.2below the bin size for v increas by a factor 2for each concutive bin.)
[In [5],for the square lattice and assumption (1),two exponents are fitted onto the data:One for the distri-bution after the first stage,and one for the distribution during the cond change.When assumption (1)is gener-alized from ∝n 2to ∝n x ,then the first exponent depends
strongly on x (which can be explained by a simple ana-lytical relation between this exponent and x ),while the cond exponent depends much less on x .]2.2Network version
In this model,we first create a network of interacting nodes by using the basic Barabasi-Albert prescription.We
雷锋纪念日
126The European Physical Journal B
fix an initial number of nodes,each one connected to the others.In the prent work,the minimum number of con-nections of a node is m=5.So,in the beginning of the simulation we have6nodes,in order that each one can be connected to the5others.After that,more and more nodes are added to the network.A new node has a proba-bility to be connected to a previous node proportional to the number of nodes that are already connected to this previous node.Thus the growth probability at any exist-ing node is proportional to the number of nodes already connected to it.We will no longer need assumption(1) and have replaced this assumption and the regular lattice by the Barabasi network without such an assumption.
After preparing the network,we start with the election process,which is now different from that in the previous three-dimensional model.Thefirst step is the distribution of candidates.Again,the state of a node,that means,the value n of a node on the network,reprents that this voter has given the preference to that candidate n.Thou-sand candidates are distributed at random,disregarding the number of connections of a we pick a node at random from the half million nodes to which we let the network grow,and then a candidate at random.Now,the campaign starts.At each time step we visit all the nodes. For each node,we have the following process:
•If a node i has already lected preference for a candi-date,we choo a connected node j at rando
m.If node
i has no candidate(n=0),we go to another randomly
lected node.
•If node j has the same candidate as node i,they try to convince all the nodes connected with them.The probability to convince others for each of the two nodes is now inverly proportional to the time-independent numbers of nodes connected with it,meaning that each node convinces–on average–one other node at each process.
•If node j has no candidate,node i tries to convince it to accept its own candidate,with the same probability as described above.
•If node j has a different candidate from node i,we skip to another node i.
Again,as described above for the3d version,we do not wait for a equilibrium state.It is important to men-tion that,different from a square lattice,where an equilib-rium state is reached in a time proportional to the number of sites,in the network it is reached rapidly,after about 102iterations.In both cas,infinal equilibrium all the sites have the same state.
Figure2shows that again except for the smallest and the largest numbers v of votes,the hyperbolic law(2)is obeyed well at intermediate times t=40.
3Summary
Whether we simulate the election process on a square lat-tice,a simple cubic lattice or a Barabasi network,we re-cover the same hyperbolic law as found in real elections.Our simulations on a Barabasi network have the advan-tage that we no longer need assumption(1)for the purpo of getting a realistic vote distribution with decay expo-nent1in the center.
Our study has shown that the hyperbolic law obrved empirically is a rather robust conquence of our modi-fid Sznajd model,since we found itfirst on the square lattice[5]and now on both the cubic lattice and the Barabasi network.Either we u a regular lattice and as-sumption(1),or we u the Barabasi network without assumption(1);thefinal results are similar.The fact that the hyperbolic law is obrvable on the Barabasi network, which is a more realistic model of social interactions than the lattices,provides evidence that the Sznajd model may well capture important aspects of the voting mechanism. The advantage of using the Barabasi network instead of regular lattices is not only that it is more realistic but also that we can drop assumption(1)which is a kind offine tuning the system to
criticality.Of cour,the Barabasi model,and the related assumptions are also rules(as we have rules when constructing a lattice too)but the differ-ence is similar to what we have for usual and lf-organized criticality:Assumption(1)puts in some exponent at the beginning,through(n/N tot)2,on which thefinal expo-nent depends somewhat(e end of Sect.2.1),while the Barabasi growth process leads by itlf to a power-law distribution of the number of connections for a node,and combined with the Sznajd model gives the desired vote distribution with its intermediate power law.
Moreover,rule(1)was introduced ad hoc to explain the election results,while the rules of[14]were stated before,independent of the prent application.We are not aware of other voter models[16]explaining the hyperbolic law found empirically in[3].
We thank Ana Proykova for suggesting the3d simulation.ATB acknowledges the hospitality of the Institute for Theoretical Physics from the University of Cologne.This work was par-tially supported by the Brazilian Agencies CNPq,FINEP and by the Hungarian OTKA T029985.
References
1.Exotic Statistical Physics,edited by A.P¸e kalski,K.
Sznajd-Weron,Physica A285,1–238(2000).
2. C.Bordogna, E.Albano,Phys.Rev.Lett.87,118701
(2001).
3.R.N.Costa-Filho,M.P.Almeida,J.S.Andrade Jr.,J.E.
Moreira,Phys.Rev.E60,1067(1999).
4. D.Stauffer,A.Aharony,Introduction to Percolation The-
ory(Taylor and Francis,London,1994); A.Bunde,S.
Havlin,Fractals and Disordered Systems(Springer,Berlin-Heidelberg1996);M.Sahimi,Applications of Percolation Theory(Taylor and Francis,London,1994).
A.T.Bernardes et al.:Election results and the Sznajd model on Barabasi network127
5. A.T.Bernardes,U.M.S.Costa,A.D.Araujo,D.Stauffer,
Int.J.Mod.Phys.C12,159(2001).
6.J.A.Ho l yst,K.Kacperski,F.Schweitzer,Annual Reviews
of Computational Physics IX(World Scientific,Singapore 2001),p.275;K.Kacperski,J.A.Ho l yst,Physica A287, 631(2000).
7.K.Sznajd-Weron,J.Sznajd,Int.J.Mod.Phys.C11,1157
(2000).
8. D.Stauffer,A.O.Sousa,S.Moss de Oliveira,Int.J.Mod.
Phys.C11,1239(2000).
9.K.Sznajd-Weron,R.Weron,Int.J.Mod.Phys.C13,
No.1(2002);R.Ochrombel,Int.J.Mod.Phys.C12,1091 (2001);I.Chang,Int.J.Mod.Phys.C12,No.10(2001);
A.S.Elgazzar,Int.J.Mod.Phys.C12,No.10(2001).
10. A.A.Moreira,J.S.Andrade Jr.,D.Stauffer,Int.J.Mod.
Phys.C12,39(2001).11.J.Adler,Physica A171,453(1991).
12. D.Stauffer,J.Phys.A24,909(1991).
13.T.B.Liverpool,Physica A224,589(1996).
14. A.L.Barabasi,R.Albert,Science286,509(1999).
15.R.Albert,H.Jeong, A.L.Barab´a si,Nature401,130
(1999);S.Redner,Eur.Phys.J.B4,131(1998);
A.L.Barab´a si d-mat/0104162;R.Albert,A.L.
Barab´a si,Rev.Mod.Phys.(in press),cond-mat/0106096.
For“small world”effect in social interactions e S.
Milgram,Psychology Today1,61(1967).
16.For recent work I.Dornic,H.Chate,J.Chave,
H.Hinrichn,Phys.Rev.Lett.87,045701(2001);R.
Heglmann,talk atfifth conference on Simulating Soci-ety,Kazimierz Dolny,Sept.2001;G.Deffuant,D.Neau,F.
Amblard,G.Weisbuch,Adv.Complex Syst.3,87(2000).
