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Weighted Scale-Free Networks with Stochastic Weight Assignments

Weighted Scale-Free Networks with Stochastic Weight Assignments
Weighted Scale-Free Networks with Stochastic Weight Assignments

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Weighted Scale-Free Networks with Stochastic Weight

Assignments Dafang Zheng 1,?,Ste?en Trimper 1,Bo Zheng 1,2and P.M.Hui 31Fachbereich Physik,Martin-Luther-Universit¨a t,D-06099Halle,Germany 2Department of Physics,Zhejiang University,Hangzhou 310027,P.R.China 3Department of Physics,The Chinese University of Hong Kong,Shatin,New Territories,Hong Kong Abstract We propose and study a model of weighted scale-free networks incorpo-rating a stochastic scheme for weight assignments to the links,taking into account both the popularity and ?tness of a node.As the network grows the weights of links are driven either by the connectivity with probability p or by the ?tness with probability 1?p .Results of numerical simulations show that the total weight associated with a selected node exhibits a power law distribution with an exponent σ,the value of which depends on the proba-bility p .The exponent σdecreases continuously as p increases.For p =0,the total weight distribution displays the same scaling behavior as that of the connectivity distribution with σ=γ=3,where γis the exponent charac-terizing the connectivity distribution.An analytical expression for the total weight is derived so as to explain the features observed in the numerical re-

sults.Numerical results are also presented for a generalized model with a

?tness-dependent link formation mechanism.

PACS numbers:89.75.Hc,05.65.+b,02.50Cw,84.35.+i

Typeset using REVT E X

I.INTRODUCTION

Many complex systems,including social,biological,physical,economic,and computer systems,can be studied using network models in which the nodes represent the constituents and links or edges represent the interactions between constituents[1,2].One of important measures of the topological structure of a network is its connectivity distribution P(k),which is de?ned as the probability that a randomly selected node has exactly k edges.In traditional random graphs[3,4]as well as in the small-world networks[5–7]the connectivity distribution shows exponential decay in the tail.However,empirical studies on many real networks showed that the connectivity distribution exhibits a power law behavior P(k)~k?γfor large k[1,2].Networks with power-law connectivity distributions are called scale-free(SF) networks.Typical examples of SF networks include the Internet[8–10],World-Wide-Web [11–13],scienti?c citations[14],cells[15,16],the web of actors[17],and the web of human sexual contacts[18].The?rst model of SF networks was proposed by Barab′a si and Albert (BA)[19].In BA networks two important ingredients are included in order to obtain power law behavior in the connectivity distributions,namely the networks are continuously growing by adding in new nodes as time evolves,and the newly added nodes are preferentially attached to the highly connected nodes.The idea of incorporating preferential attachment in a growing network has led to proposals of a considerable number of models of SF networks [20–26](see also Refs.[1,2]and references therein).

In most growing network models,all the links are considered equivalent.However,many real systems display di?erent interaction strengths between nodes.It has been shown that in systems such as the social acquaintance network[27],the web of scientists with collaborations [28],and ecosystems[29],links between nodes may be di?erent in their in?uence and the so-called the weak links play an important role in governing the network’s functions.Therefore, real systems are best described by weighted growing networks with non-uniform strengths of the links.Only recently,a class of models of weighted growing networks was proposed by Yook,Jeong and Barab′a si(YJB)[30].In the basic weighted scale-free(WSF)model of YJB,

both the topology and the weight are driven by the connectivity according to the preferential attachment rule as the network grows.It was found that the total weight distribution follows a power law P(w)~w?σ,with an exponentσdi?erent from the connectivity exponentγ.It was also shown analytically that the di?erent scaling behavior in the weight and connectivity distributions are results of strong logarithmic corrections,and asymptotically(i.e.,in the long time limit)the weighted and un-weighted models are identical[30].

In real systems one would expect that a link’s weight and/or the growth rate in the number of links of a node depend not only on the“popularity”of the node represented by the connectivity,but also on some intrinsic quality of the node.The intrinsic quality can be collectively represented by a parameter referred to as the“?tness”[31,32].Besides popularity,the competitiveness of a node in a network may depend,taking for example a node being an individual in a certain community,on the personality,survival skills,character, etc..A newly added node may take into account of these factors beside popularity in their decision on making connections with existing nodes and on the importance of each of the established links.Clearly,there is always a spectrum of personality among the nodes and therefore a distribution in the?tness.While one may argue that factors determining the popularity may overlap with those in?tness,it is not uncommon that popularity is not the major factor on the importance of a connection.For example,we often hear that a popular person actually has very few good friends,and an in?uential and powerful?gure in a network may often be someone very di?cult to work with.In the present work,we generalize the WSF model of YJB to study the e?ects of?tness.In our model,the weights assigned to the newly added links are determined stochastically either by the connectivity with probability p or by the?tness of nodes with probability1?p.The scaling behavior of the total weight distribution is found to be highly sensitive to the weight assignment mechanism through the parameter p.

The plan of the paper is as follows.In Sec.II,we present our model and simulation results.In Sec.III,we derive an analytical expression between the total weight and the total connectivity of a node and provide a theoretical explanation on the features observed in the

numerical results.Results on a generalized model with a?tness-dependent link formation mechanism are presented in Sec.IV,together with a summary of results.

II.THE MODEL AND NUMERICAL RESULTS

The topological structure of our model follows that of the BA model of SF networks[19].

A small number(m0)of nodes are created initially.At each time step,a new node j with m(m≤m0)links is added to the network.These m links will connect to m pre-existing nodes in the system according to the preferential attachment rule that the probabilityΠi of an existing node i being selected for connection is proportional to the total number of links k i that node i carries,i.e.,

k i

Πi=

{i′}k i′,(2) where {i′}is a sum over the m nodes to which the new node j is connected.With probability 1?p,w ji is determined by the?tness through

ηi

w ji=

In Eqs.(2)and(3),w ji is normalized so that the sum of the weights for the m new links is unity,i.e., {i′}w ji′=1[30].For p=1,our model reduces to the basic WSF model of

YJB in which the weights are driven by the connectivity alone[30].For p=0,the weights are driven entirely by the?tness of the nodes.For0

We performed extensive numerical simulations on the model.In our simulations,we studied networks up to N=5×105nodes with m=m0=5.For each value of p,results are obtained by averaging over10independent runs.First we study the total weight distribution P(w),which is de?ned as the probability that a randomly selected node has a total weight w.The total weight of a node i is given by the sum of the weights of all links connected to it,i.e.,w i= j w ij.Figure1shows that P(w)behaves as a power law P(w)~w?σ, with an exponentσthat decreases from the value of3at p=0continuously as p increases. For p=1,σ=2.4,a result in agreement with that of the WSF model of YJB[30].For p=0,σ=3(=γ)showing that P(w)follows the same scaling behavior as P(k).YJB found that the scaling behavior of P(w)depends strongly on m[30]in their model.In the present model,we found that the m-dependence persists for all p>0.Only when p=0,σbecomes independent of m.

It is also interesting to study the dynamical behavior of the total weight w i(ηi,t)of some node i with?tnessηi.Fig.2shows that w i(ηi,t)grows with a power law behavior with time with an exponentδthat depends on p.For p>0,δ>β,whereβ=1/2is the exponent characterizing the dynamical behavior of the connectivity k i(t)[19].For p=0,w i(ηi,t) shows the same scaling behavior as k i(t)withδ=β=1/2.For0

The probability distribution P(w ij)of the weights w ij of individual links is also worth investigating.To suppress statistical?uctuations,Fig.3shows the cumulative distribution,

P(x>w ij),instead of P(w ij),on a log-linear scale.For p=0,P(x>w ij)decays expo-nentially in the tail.Recall that P(w)and w i(ηi,t)show identical behavior as P(k)and k i(t)for p=0respectively,and the latter two quantities are not sensitive to the weight assignment scheme.Here P(x>w ij)shows an exponentially decaying behavior,implying that the weighted and un-weighted models are not totally identical even for p=0.For p>0,the tail deviates from an exponential decaying form and decays faster as p increases. For p=1,we recover the results in the YJB model[30].

III.ANALYTICAL SOLUTION

To understand the di?erent behavior between w i(ηi,t)and k i(t)(as well as between P(w) and P(k))found in numerical simulations,we derive an analytical expression for the total weight w i(ηi,t)of a node i with?tnessηi at time t.Following YJB[30],w i(ηi,t)can be expressed as

w i(ηi,t)=1+ t t0i ∞m 10?P i(m,t′)w ji(ηl,k l)?(k l)ρ(ηl)dηl dk l dt′,(4) where?P i(m,t)is the probability that node i is selected for connection to a new node j at time t for given m and it is related toΠi in Eq.(1)by a factor of m.Here,t0i is the time at which the node i has been added to the system.w ji(ηl,k l)is the weight assigned to the link between node j and node i.?(k)andρ(η)are the probability distributions of k andη, respectively.According to the stochastic weight assignment scheme modelled by Eqs.(2) and(3),the weight w ji(ηl,k l),on the average,can be written as

w ji(ηl,k l)=p k i

ηi+ηl

,(5)

for the simple case of m=2.Generalization to arbitrary value of m is straightforward.

From the connectedness of the SF model,?P i(m,t),?(k)and k i(t)are given by[33,30]

?P i (m,t)=mΠi=

k i(t)

k i(t)=m

t0i √

ηi ]k i(t)?

1

t0i

)2?4ln2ln t

ηi

k i(t),(10)

leading to the same scaling behavior of w i(ηi,t)and k i(t),as observed in the simulation results.For p=1corresponding to the WSF model of YJB[30],the dynamical behavior of w i(ηi,t)deviates most from that of k i(t).For arbitrary m,w i(ηi,t)follows a similar form with m dependence coming into the second term on the right hand side of Eq.(9).

IV.DISCUSSION

Our model can be easily generalized to allow for a?tness-dependent link formation mechanism[31,32].In the basic model with?tness[31],the probabilityΠi that a new link is established with an existing node i is determined jointly by the node’s connectivity k i and ?tnessηi with

Πi=

ηi k i

log k

k?γ′,(12)

withγ′=2.255.Fig.4shows the numerical results for the total weight distribution P(w)for three di?erent values of p=0,0.5and1.It is found that P(w)follows the same generalized power law form as P(k),but with a di?erent exponentσ′that depends on p.For p>0,σ′<γ′.Only for p=0,P(w)and P(k)have the same exponent ofσ′=2.25~γ′.For the cumulative distribution P(x>w ij)of weights of individual links,the numerical results are similar to those shown in Fig.3.

In summary,we proposed and studied a model of weighted scale-free networks in which the weights assigned to links as the network grows are stochastically determined by the connectivity of nodes with probability p and by the?tness of nodes with probability1?p. The model leads to a power law probability distribution for the total weight characterized by an exponentσthat is highly sensitive to the probability p.If the weight is driven solely by the?tness,i.e.,p=0,P(w)follows the same scaling behavior of P(k)with the same exponentσ=γ.Similar results were also found in a generalized model with a?tness dependent link formation mechanism.An expression relating the total weight and the total connectivity of a node was derived analytically.The analytical result was used to explain the features observed in the results of numerical simulations.In closing,we note that although the total weight distribution P(w)and the connectivity distribution P(k)carry di?erent exponentsσandγfor p>0in our model,P(w)still follows a power law,i.e.,has the same functional form as P(k).The same feature was also found in the generalized model. However,one would expect that in some complex real systems even the functional forms of P(w)and P(k)may be di?erent.It remains a challenge to introduce simple and yet non-trivial models that give one behavior for the geometrical connection among the constituents and another behavior for the extend of connectivity between constituents.

ACKNOWLEDGMENTS

This work was supported by a DFG grant TR3000/3-3.One of us(P.M.H.)acknowledges the support from the Research Grants Council of the Hong Kong SAR Government under

grant number CUHK4241/01P.

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FIGURE CAPTIONS

Figure1:The weight distribution P(w)as a function of the total weight w on a log-log scale for di?erent values of p=0,0.1,0.5,1.0.The two solid lines are guide to the eye corresponding to the exponentsσ=2.4and3.0,respectively.

Figure2:The total weight w i(ηi,t)of a randomly selected node i with?tnessηi(=0.75)as a function of time t on a log-log scale for di?erent values of p=0,0.5,1.0.The solid line is a guide to the eye corresponding to an exponentσ=0.5.

Figure3:The cumulative distribution P(x>w ij)of the weights of individual links as a function of w ij on a log-linear plot for di?erent values of p=0,0.1,0.5,1.0.

Figure4:The weight distribution P(w)as a function of the total weight w on a log-log scale for di?erent values of p=0,0.5,1.0in a model with?tness-dependent link formation mechanism.The two solid lines are plotted according to the form of Eq.(12),but with an exponentσ′characterizing P(w)that takes on the values1.82and2.25respectively.

w 100101102103104

P (w )10-10

10-9

10-8

10-7

10-6

10-5

10-410-3

10-2

10-1

100

p=0p=0.1p=0.5p=1.0

-3.0-2.4Figure 1

Zheng et al.

t 104105106

w i (ηi ,t )

101

102

103104

p=1.0p=0.5p=0

0.5

Figure 2

Zheng et al.

W ij 0.0.2.4.6.8 1.0

P (x >w i j )10-4

10-3

10-210-1

100

p=0

p=0.1

p=0.5

p=1.0

Figure 3

Zheng et al.

w 100101102103104

P (w )10-9

10-8

10-7

10-6

10-5

10-4

10-310-2

10-1

100

p=0p=0.5p=1.0

-2.25-1.82Figure 4Zheng et al.

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2、 With the meal over , we all went home.(with+名词+副词,作时间状语) 3、The master was walking up and down with the ruler under his arm。(with+名词+介词短语,作伴随状语。) The teacher entered the classroom with a book in his hand. 4、He lay in the dark empty house,with not a man ,woman or child to say he was kind to me.(with+名词+不定式,作伴随状语)He could not finish it without me to help him.(without+代词 +不定式,作条件状语) 5、She fell asleep with the light burning.(with+名词+现在分词,作伴随状语) Without anything left in the with结构是许多英 语复合结构中最常用的一种。学好它对学好复合宾语结构、不定式复合结构、动名词复合结构和独立主格结构均能起很重要的作用。本文就此的构成、特点及用法等作一较全面阐述,以帮助同学们掌握这一重要的语法知识。 二、with结构的用法 with是介词,其意义颇多,一时难掌握。为帮助大家理清头绪,以教材中的句子为例,进行分类,并配以简单的解释。在句子中with结构多数充当状语,表示行为方式,伴随情况、时间、原因或条件(详见上述例句)。 1.带着,牵着…… (表动作特征)。如: Run with the kite like this.

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适应,精神和心情就会受到一定的影响,大脑控制着人的精神世界, 有可能促发精神分裂症。 5、身体方面:细菌感染、出现中毒情况、大脑外伤、肿瘤、身体的代谢及营养不良等均可能导致使精神分裂症,身体受到外界环境的 影响受到一定程度的伤害,心里受到打击,无法承受伤害造成的痛苦,可能会出现精神的问题。 对于精神分裂症一定要配合治疗,接受全面正确的治疗,最好的 疗法就是中医疗法加心理疗法。早发现并及时治疗并且科学合理的治疗,不要相信迷信,要去正规的医院接受合理的治疗,接受正确的治 疗按照医生的要求对症下药,配合医生和家人,给病人创造一个良好 的治疗环境,对于该病的康复和痊愈会起到意想不到的效果。

(完整版)with的复合结构用法及练习

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With复合结构的用法小结

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with的复合结构用法及练习

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