The Study on a Comprehensive Evolving Model of Directed Weighted Stock Network Based on

Information Services

Qi Jie

Dept. of Management Science School of Management, Xi’an Jiaotong University

Xi’an 710049, China e-mail：hello_xiaoqi@stu.xjtu.edu.cn

Hu Ping, Wang Bingqing Dept. of Management Science

School of Management, Xi’an Jiaotong University Xi’an 710049, China

e-mail：helenhu@xjtu.edu.cn

Abstract—Based on correlation of the stock price fluctuation

of the information services stock network in Shanghai A-share and Shenzhen A-share, this paper established a stock network according to the given weight of the company’s relative value. Then it proposes a comprehensive directed weighted evolving network model that integrated fitness-based preferential attachment, node deletion, link establishment and link deletion. Next, after a deduction about the evolution of the comprehensive network model, it concludes the power-law feature of strength distribution. Finally, according to the distribution of different node fitness, the paper analyzes the evolution of the stock network and verifies it by using numerical simulations.

Keywords—stock network ； evolution model ； power-law feature；information service industry

I. INTRODUCTION

Recently, much research has been done on the formation and evolution of complex network in the academic circle by widely using in nature, engineering and social world [1]. Meanwhile, scholars home and abroad have done some studies on the application of complex network theory of the important global stock market, such as Hong Kong's Hang Seng[2], Italian stock market[3], Shanghai and Shenzhen stock market[4][5], mainly to explore the small world, scale-free feature, and evolution mechanism. Despite the in-depth studies, most domestic research has been limited to model of undirected and weighted stock network, lacking mechanism of node addition, node deletion, link establishment and link deletion, which makes model much simpler than stock network in reality. The paper proposes a comprehensive model of stock network evolution of information service industry, considering fitness of nodes, weight of link assigned by probability distribution, mechanism of node addition, node deletion, strengthening and weakening of the relationship between stocks. Finally, the paper makes a simulated model and tests it according to the data.

II. LITERATURE REVIEWS Basic mechanism of network evolution is node addition,

node deletion, edge establishment and edge deletion. WS

model, proposed by scholars Watts and Strogtz in 1998, is a small world network model [6] which has higher level of clustering and smaller network diameter, concerning the random process of the establishment of links. BA classic network is the network model including two most important evolution mechanisms - network continuous growth and the new node preferential attachment [7], which was put forward by Albert - Laszlo Barabasi and Reka Albert in 1999. They think nodes having higher connectivity are more likely to grow, which provokes new thoughts about node addition and edge connection. Based on the BA model, Ginestra Bianconi and Albert - Laszlo Barabasi [8] first proposed the Fitness model adding node fitness for the study on node growth and link establishment, considering the node's own power. Later, Joseph S.Kong and Vwani P Roychowdhury [8] proposed double preferential attachment (DPA) mechanism [10][11] and node optimizing survival mechanism based on BA model. These network evolution mechanisms play an important role in the research and analysis of real network’s characteristics, which are also taken into consideration while studying the evolution of stock network in this paper.

While network evolution factors mentioned above have been discussed in an undirected network model, they are seldom explored in the directed weighted model. Scholars SH Yook, H Jeong and AL Barabasi proposed directed weighted model, including mainly two new models--WE model and WSF model[12]. The model supposed that the weight and degree of node are in positive correlation, changing the binary of weight. This weight put forward the important thoughts about weights in the network. Dafang Zheng [13] built a scale-free and weighted network model combining with random weight distribution mechanism while considering the node's popularity and fitness. The weight of stock network’s edge, which can not be fixed, is related to the fluctuation of stock price. Thus it is more suitable to adopt a weight distribution method based on certain probability. In addition, when it comes to directed weighted network, link deletion is seldom considered. Taken the study object of this paper into consideration, link deletion is a common evolution behavior in the stock network of information service industry, so it is necessary to continue the study.

This work was supported in part by the National Natural Social ScienceFoundation of China under grants 10BGL032, and the electronic andinformation department of the Chinese academy of Engineering in 2013(2013-02-XY-003）.

978-1-4673-4843-0/13/$31.00 © 2013 IEEE

Pointing the stock network, in 2001, Kim [6] and other authors put forward the stock network model based on 500 standard listed companies. This model, taking the stock as the node, using the correlation coefficient of the logarithmic of closing price return in a day, regarding the strength between nodes as the weight of link, finally proved that the network was a scale-free network. Based on the foundation of Kim’s work, Nitin [14] and some scholars found that correlation coefficient of price fluctuation approximately obeyed normal distribution and concentrated in the interval (0.5, 0.5), eliminating interest rate, inflation, exchange rate and the interference of outside factors. Domestic scholar Huang Weiqiang [15] built a network model with price fluctuation on the basis of 1080 stocks of Shanghai and Shenzhen Stock Market. He processed the threshold and analyzed the topology and characteristics of the network. This paper follows Kim’s thought to build a stock network, and chooses stocks of information service industry in Shanghai and Shenzhen stock market, further exploring the dynamic characteristics of evolving model of stock network.

III. THE PROPOSAL OF EVOLUTION MECHANISM OF STOCK NETWORK

Combine strength distribution, weight distribution, node addition, link establishment and link deletion to build the evolution mechanism of stock network as follows:

Initial condition: there are m0 nodes and e0 edges.

First step: node addition. Add a new node j at every step with fitness λi from fitness distribution ρ(λ).

Second step: Preferential Attachment. m edges which is brought by the new node j will connet to the existing nodes. The probability of the existing node i being connected is supposed to be ∏i, the definition is:

i ii

h h h

ss

λλ

Π =∑ （1）

Third step: weight distribution. After new node j connects to the existing node i, assign weight ωij=ω according to the probability p, and w follows distribution ρ(ω ). Otherwise, assign weight ωij=0 with probability 1-p.

Forth step: DPA. am(a≥0）edges will be added with the selection mechanism of both ends based on Πi as (1).

Fifth step: edge deletion. Delete bm(b≥0)edges randomly among existing edges.

Sixth step: node deletion. Delete an existing node and its all links with probability c.

A. Model Theorical Derivation This paper adopts continuum theory [16][17]to analyze the

dynamic features of evolving model of stock network, based on the mechanism above.

Suppose that if the node i exists at time t, the expectation of the strength rate at time t is:

( , ) ( , )( , ) ( , ) ( , )[ ( )] 2 [ ( )]( ) ( ) ( )/2 ( )

i is i t s i ts i t s i t s i tm pE am pE bm ct Rt Rt Gt N t

λ λω ω∂ = + − −∂ （2）

R(t) stands for the sum of products of all the fitness and strength, D(i,t)stands for the probability of node i still existing in the network at time t, the expression of R(t) is:

0( ) ( , ) ( , )

t

iR t D i t s i t diλ= ∫ （3）

G(t) stands for the sum of degrees of all nodes in the network at the time t, that is:

0( ) ( , ) ( , )

tG t D i t k i t di= ∫ （4）

N(t) stands for the expectation of the number of the nodes in the net at the time t, that is:

( ) (1 )N t c t= − （5）

According to the definition of D(i,t), it must satisfy the following differential form:

( , 1) ( , )[1 ]( )cD i t D i t

N t+ = −

（6）

Then,

( , ) ( , ) ( , )( ) 1

D i t cD i t c D i tt N t c t

∂ = − = −∂ − （7）

Solve the differential equation (7) combining boundary condition, that is:

1( , ) ( )c

ctD i ti

−= （8）

B. Self-consistency Verification According to Nima’s calculation, based on the mechanism

of node addition, preferential attachment, node deletion, link establishment and link deletion, the connectivity rate equation of undirected and unweight network is:

( , ) ( , ) ( , ) ( , ) ( , )2( ) ( ) ( ) / 2 ( )

k i t k i t k i t k i t k i tm am bm ct G t G t G t N t

∂ = + − −∂ （9）

Multiply by D(i,t) on both side of (9) and compute the integral of both side from 0 to t, we can get:

0 00

( , ) ( , ) ( , ) ( , )( , )( , ) (1 2 2 )( ) ( )

( )(1 2 2 )(1 )

t tt D i t k i t di D i t k i t dik i tD i t di a b m c

t G t N tG ta b m c

c t

∂ ∂∂ = + − −∂

= + − −−

∫ ∫∫

(10)

Combining the condition, k(t,t)=m, D(t,t)=1， we can get expression from integrating (10):

0 0 0

( , ) ( , )( , ) { ( , ) ( , )} ( , )

( ) ( )( 1)

t t tk i t Di tDi t di Di t k i t di k i t dit t t

Gt cm Gtt t c

∂ ∂ ∂= −∂ ∂ ∂

∂= − −∂ −

∫ ∫ ∫ (11)

Then,

( ) ( )( ) (1 2 2 )( 1) (1 )

G t c G tm G t a b m ct t c c t

∂ − − = + − −∂ − − (12)

To solve the differential equation, get:

1( ) (2 2 2 )1

cG t a b mt mtc

−= + − = Ψ+ (13)

And 1(2 2 2 )1

ca bc

−Ψ = + −+ （14）

Now that (2) is hard to solve, R(t) can not be got. This paper uses Sarshar’s[19] method which adopts self-consistent nature to verification. The main process is :first suppose that s(i,t) follows power law, namely, s(i,t)~tβ,andβ is the growth coefficient of the node. Then, solve the equation and prove the hypothesis.

Thus，the suppose can be got:

( )( , ) ( ) its i t mi

β λ= （15）

Here, β(λi) means that the growth coefficient of the node depends on the fitnessλi. Then, calculate R(t) by putting (15) to (13),

( )

1

( )1

( ) ( ){ ( , ) ( ) }

( )1 [ ( ) ]

1

t

cc

tR t D i t m di di

t tm dcc

β λ

β λ

λρ λ λ

λρ λ λβ λ

−−

=

−=− −

−

∫ ∫

∫

（16）

Because the growth coefficient is not bigger than

1,( )

1lim 0cc

tt

β λ −−

→∞→

and (11) is simplified to

R(t) ≈Φmt （17）

and

( )

1 [ ( ) ]1

dcc

λρ λ λβ λ

Φ =− −

−

∫ （18）

Put (5) (13) (17) to (2), we can get

(2 1) ( , )( , ) 2 ( , ) ( , )( )(1 )

( , ) ( )

i

i

a m s i ts i t bms i t cs i tpEt mt mt c t

s i t u wt

λ ω

λ

+∂ = − −∂ Φ Ψ −

= −Φ

（19）

Where (2 1) ( )u a pE ω= + （20）

And

21 (1 )(1 )

b c b c acwc c a b

+ += + =Ψ − − + + （21）

The form of the solution of differential equation （19）must correspond to (15), then

( ) u wλβ λ = −

Φ （22）

With this, we prove the self-consistency, which displays that s(i,t)obeys power-law and its exponent is decided by (22) and effected by fitness λ.

C. The Confirmation of The Strength Distribution Afterβis confirmed, based on Sarshar’s research, whenλis

confirmed, every λ will have a fixed power-law exponent, we get:

1( ) 1(1 ) ( )c

γ λβ λ

= +− （23）

According to the calculation of Binaconi, the strength distribution p(s) of this model actually consists of various values of β(λ), and the expression is:

max

min

( )( ) ( )P s s dλ γ λ

λρ λ λ−∝ ∫ （24）

Combining with (22) (23), we get:

max

min

[1 ](1 )( )( ) ( ) c u wP s s d

λ λλ

ρ λ λΦ− +

− − Φ∝ ∫ （25）

According to Binaconi’s research, it is strongly influenced by maximum λ.

IV. MODEL ANALYSIS After getting the mathematical derivation of the evolution

model of the stock network, we will calculate the model in the definite cases which are node fitness homogeneous, node fitness obeying (0,1) uniform distribution, and node fitness obeying exponential distribution to analyze the strength distribution evolution ,then compare the simulation with mathematical derivation to prove the conclusion.

A. Node Fitness Homogeneous We first discuss the simplest situation which is node fitness

homogeneous-the competition of all nodes is the same. Let us assume that λ=1. According to the definition of probability distribution, p(λ)is infinity with λ=1. Therefore (18) is hard to solve. Now, this paper used Dirac Delta Function which is an illusion function with special points on X-axis whose probability density is infinity. It rules:

( ) ( ) ( )f t t T dt f Tδ+∞

−∞− =∫ (26)

where T is special point and corresponds to the value of λ. Hence, (13) can be simplified:

max

min

( ) 1

1 ( ) 11 1

dc u cu w wc c

λ

λ

ρ λ λλΦ= =− − + − + +

Φ − Φ −

∫ (27)

Then, we can get:

1

11

ucw

c

+Φ =+ +

− (28)

Substitute (28) to (22), we get:

(1 )1 1( )1

cu wcu w w

uβ λ

+ +−= − = −

Φ + (29)

Substitute (29) to (23), we get:

1 1( ) 1 1(1 ) ( ) (1 )

uc u w c

γ λβ λ

+= + = +− − − (30)

Substitute (20) (21) to (30), we integrate:

1 (2 1) ( )( ) 1(2 1) ( )

1

a pEb c aca pE

a b

ωγ λω

+ += + + ++ −+ − (31)

If a=0, b=0, c=0, p=1, E(ω)=1, it ignores weight ,direction ,node deletion and link establishment, which degenerates to classical BA Model and gets γ=3 corresponding to the results of Barabasi’s research[18]. This result proves that our model is right. However, BA model is a special case at extreme condition. Keeping other condition constant, the inequality

/ ( ) 0Eγ ω∂ ∂ < illustrates γ(λ) and E(ω)have monotone decreasing relationship. Similarly, if inequality is / 0pγ∂ ∂ > , γ(λ) and p have monotone decreasing relationship.

If b=0, a=0, there is no link establishment and link deletion. Keeping other condition constant, γ(λ) increases as deletion probability c increases. Otherwise, γ(λ) decreases. Those means strength distribution exponent γ(λ) and node deletion probability c have Monotone increasing relationship.

If c=0、b=0, there is no deletion mechanism. Keeping other condition constant, there is an inequality / 0aγ∂ ∂ < which illustrates that strength distribution exponent and new link coefficientαhave monotone decreasing relationship and satisfies the inequality2 2 1 / (2 1) ( )a pEγ ω< ≤ + + .

The simulation results for proving our theory are reported in Fig.1.Control groups (a) and (b) verify that strength distribution exponent γ(λ) and weight probability p have monotone increasing relationship since the absolute value of slope of picture(b) is bigger than that in picture(a) when set different values of p from 0.4 to 0.8. Control groups (c) and (d) verify that strength distribution γ(λ) and node deletion probability c have monotone increasing relationship without mechanism of link establishment and link deletion. Without deletion mechanism, when set the value ofαfrom 0.3 to1, strength distribution exponent γ(λ) and new link coefficientαhave monotone decreasing relationship. The figure is very similar with above ones so that we omit the simulation results here.

(a)a=1,b=1,c=0.3,p=0.4

(b)a=1,b=1,c=0.3,p=0.8

(c)a=0,b=0,c=0.3,p=0.4

(d)a=0,b=0,c=0.7,p=0.4

Fig. 1. The degree distribution of node obeying fitness homogeneous network.

B. Node Fitness Obeying (0,1) Uniform Distribution In reality, nodes obey uniform distribution, which means

node fitness nearly appears in the same probability in an interval. Hence, when node fitness follows (0, 1) uniform distribution, we get (26). Then combining (22), we observe (18):

1

01

1

du cw

c

λ λλ

Φ =⎛ ⎞− − −⎜ ⎟Φ −⎝ ⎠

∫ (32)

Simplify the above equation and we get: 2

ln z u uz u z

+=− (33)

Where (1 )

1cz w

c= + + Φ

+ (34)

Observing (33), analytical relation between z and u is hard to solve, so this paper uses Matlab R2009b to obtain its numerical solution.

The fitting curve of u and z approximates into linear distribution. We apply linear regression analysis with Spss19.0 and get:

0.866 0.413z u= + (35)

Insert (35) back into (34) and get: 0 .8 6 6 0 .4 1 3

11

ucw

c

+Φ =+ +

− (36)

Insert (36) back into(22) and get:

(1 )1( )

0.866 0.413

cu wcu w w

uλβ λ

+ +−= − = −

Φ + (37)

Substitute above equation to (23) and we get:

0.866 0.413( ) 1[(1 )(1 ) ] (1 )(0.866 0.413)

uu c w c w c u

γ λ += +− + + − − + (38)

Substitute (20), (21) to (38), and integrate it:

(1 )[0.866(2 1) ( ) 0.413]( ) 1(2 1) ( )(1 ) ( )[0.866(2 1) ( ) 0.413]

a b a pEa pE a c ac b c ac a pE

ωγ λω ω

+ − + += ++ + + + − + + + + (39)

If we don’t consider weight, direction, node deletion, link addition and link deletion, namely, a=0、b=0、c=0、p=1，E(ω)=1, we calculate that γ is 2.279. In classic Fitness model, Bianconi and other authors calculate power law exponent γ is 2.255 under fitness obeying [0,1] uniform distribution. Obviously, Comparing two results γ, this model’s result approximates the classic model with only 1.05% error which is in the normal scale considering that numerical solution is used many times. Therefore, our model is correct.

If there is no link addition and link deletion, namely, a=0, b=0 and others hold fix, when 0.134 ( ) 0.413 0pE ω − > and

/ 0cγ∂ ∂ < , power law exponent γ(λ) and node deletion

probability c have monotone decreasing relationship. Similarly, when0.134 ( ) 0.413 0pE ω − < , the conclusion is opposite.

(a)a=0,b=0,c=0.3,p=0.4

(b)a=0,b=0,c=0.7,p=0.4

(c)a=0.3,b=0,c=0,p=0.4

(d)a=1,b=0,c=0,p=0.4

Fig. 2. The strength distribution of node fitness obeying [0,1] uniform distribution.

If there is no deletion mechanism, namely, b=0,c=0 and other conditions keep fixing, when / 0aγ∂ ∂ < , power law

exponent γ(λ) and link establishment coefficient α have monotone decreasing relationship and 1.866 1.866 0.413 / (2 1) ( )a pEγ ω< ≤ + + . Likewise, we can analyze that the relationship of γ(λ) and p have monotone decreasing as well as the relationship of γ(λ) and E(λ).

In order to confirm the derivation results, we simulate it. In Fig 2, comparing control groups (a) (b), we can verify the power law exponent γ(λ) and node deletion probability c have monotone decreasing relationship since the absolute value of slope become smaller when there is no mechanism of establishment and deletion and stay under the condition 0.134 ( ) 0.413 0pE ω − < . Likewise, the result for control groups (c) and (d) give the proof of the monotone decreasing relationship of power law exponent γ(λ) and link establishment coefficientα.

C. Node Fitness Obeying Exponential Distribution In reality network, there is a common phenomenon that a

few individuals have very rich resources and great competitiveness, which means a few nodes have very high fitness while most of nodes have low fitness. This paper discusses the situation which assumes that node fitness obeys exponential distribution. Since the real node fitness can not be infinite, we set the scale of fitness in[0,λmax]

max max0( ) 10

h

hhee

others

λ

λ λ λρ λ−

−

⎧< <⎪= −⎨

⎪⎩ (40)

Combining with (22), we integrate (18): max

max

0

1 hhe edr u h

λλλ λ λλ

−− −=Φ −∫

(41)

Where 1

1cr w

c= + +

− (42)

Observing (42), we can not be solved directly. Suppose y=γΦ, and we use mathematic software to solve the numerical relationship of y, u, λmax and h. The solving process is like that in part B. After the numerical analysis of Matlab 2009b and regression analysis of Spss 19.0, we find that the parameter h of exponent distribution has no influence on y and get the relationship as follows.

max (1.982 1.012)y uλ= − (43)

In order to be convenient for derivation, approximate (42)

as: max (2 1)y uλ= − (44)

We obtain max (2 1)uy

r rλ −Φ = =

(45)

Combining with (22), (23), (42) and (45), we obtain:

1( ) 1 (2 1) ( )(1 )(1 ) ( )2(2 1) ( ) 1

a ba pE a c b c ac

a pE

γ λ ωω

+ −= + + + + − + ++ − (46)

Kong and other writers propose classic fitness model whose power law exponent is 2 regardless of node deletion probability under the condition that node fitness obeys interceptive index distribution. Observing (46),If we don’t consider weight, direction, node deletion, link addition and link deletion, namely, a=0、 b=0、 c=0、 p=1，E（ω）=1, this model degenerates to classic Fitness model with node mechanism and power law exponent γ equals 2 no matter how node deletion probability c is. This conclusion corresponds to Kong’s completely. Meanwhile, it also illustrates that fitness model is one form of our model in extreme condition and test our model’s correction.

If there is no link addition and link deletion, namely, a=0, b=0 and others hold fix, when 0.5 ( ) 1pE ω< < , we get / 0cγ∂ ∂ < , which means power law exponent γ(λ) and node deletion probability c have monotone decreasing relationship.

If there is no deletion mechanism, namely, b=0, c=0 and other conditions keep fixed, similarly, we get / 0aγ∂ ∂ > and power law exponent γ(λ) and link establishment coefficient α have monotone decreasing relationship . Likewise, since

/ 0pγ∂ ∂ > we can analyze that the relationship of γ(λ) and p have monotone increasing relationship. Following the above thinking, we know that strength distribution exponent γ and weight distribution average E(ω) have monotone increasing relationship.

(a)a=0,b=0,c=0.3,p=0.4

(b)a=0,b=0,c=0.7,p=0.4

Fig. 3. The strength distribution of node fitness obeying exponent distribution.

The simulation result of Figure 3 is in agreement of our theory. When we change the value of c from0.3 to 0.7, control groups (a) and (b) verify that strength distribution γ(λ) and node deletion probability c have monotone increasing relationship since the absolute value of slope become bigger without mechanism of link establishment and link deletion.

Similarly, when set control groups which are a=0.3,b=0,c=0,p=0.4 and a=1,b=0,c=0,p=0.4 respectively, the curve of distribution exponent γ(λ) and new link coefficient α is almost a line just like the above groups. They have monotone decreasing relationship. The pictures of third control groups which are a=1,b=1,c=0.3,p=0.4 and a=1,b=1,c=0.3,p=0.8 are also described like first groups, showing the strength distribution exponent γ(λ) and weight probability p have monotone increasing relationship. Since they have the similar pictures with the first groups, we omit them here.

V. CONCLUSION Based on characteristics and evolution mechanism of the

information services industry stock network, combined with the theoretical limitations of the network model, this paper establishes a comprehensive evolving model of directed weighted stock network pointing to the information services industry shares in Shanghai and Shenzhen A-share , and find the conclusions are as follows:

1) If the node fitness is homogeneity, the network strength distribution exponent and network weight probability have monotonically increasing relationship; without considering weight, direction, edges addition and deletion, and node deletion, this model degenerates to Classic BA model; without considering the link establishment and deletion, the network strength distribution exponent γ and node deletion probability c is monotonically increasing relationship; without considering deletion mechanism in the network, the strength distribution exponent γ (λ) and the increasing link coefficient α have monotonically decreasing relationship.

2) If node fitness obeys [0, 1] uniform distribution, the present model degenerates into Fitness model with fitness obeying [0, 1] uniformly distribution and the error is only 1.05%. Without considering deletion mechanism, the strength distribution exponent γ (λ) and the link increasing coefficient α have monotonically decreasing relationship; without considering link addition and deletion, and 0.134 ( ) 0.413 0pE ω − > ， the power law exponent γ (λ) probability c have monotonically decreasing relationship with the node to delete.

3) If node fitness is exponential distribution, this model degenerates into Fitness Model which exactly corresponds to Kong et al. Without considering deleting mechanism, power law exponent γ and link establishment coefficient α is monotonically decreasing relationship; without considering reconstruction and deletion mechanism, and if 0.5 ( ) 1pE ω< < , the power-law exponent γ node deletion probability c have monotonically decreasing relationship; Similarly, in other circumstances, they are monotonically increasing relationship. The strength distribution exponent γ and the weight probability p have respectively monotonically increasing relationship as well as the relationship between γ and weight distribution average E(ω) .

In this paper, we select Shanghai and Shenzhen stock market stock to build network, overcoming some limitations in the past. The future research can expand to foreign stocks, such as Hong Kong stocks, and compare domestic and foreign stock

market to get the different characteristics of the information service industry. This article also has certain shortcomings: edge deletion and node deletion mechanisms are based on the random deletion mechanisms in order to facilitate in the derivation of the model. But in reality, the optional inferior deletion is more obvious features, which should be the concern for future researches.

REFERENCES [1] Jianxiang Liu. The review of complex network and its home research

progress [J]. System Science Journal 2009(15): 31-37. (in Chinese) [2] Wang, B. H. and P. M. Hui. The distribution and scaling of fluctuations

for Hang Seng index in Hong Kong stock market[J]. The European Physical Journal B-Condensed Matter and Complex Systems, 2001, 20(4): 573-579.

[3] Wang, B. H. and P. M. Hui. The distribution and scaling of fluctuations for Hang Seng index in Hong Kong stock market[J]. The European Physical Journal B-Condensed Matter and Complex Systems, 2001, 20(4): 573-579.

[4] Xintian Zhuang, Zhifeng Min. Analysis of complex network characteristics of the Shanghai securities market[J]. Northeastern University Journal (Natural Science). 2007,33(07): 1053-1056. (in Chinese)

[5] Weiwei Lu, Zhengchun Lin. The analysis of Shanghai and Shenzhen A-share market based on complex network [J]. Science Technology and Engineering. 2009,(5): 2859-2863. (in Chinese)

[6] Watts D, Strogatz S. Collective dynamics of small-world networks[J]. Nature, 1998, 393: 440-442.

[7] Jeong H, Neda Z, Barabasi AL. Measuring preferential attachment for evolving networks[J]. Europhysics Letters, 2003, 61: 567.

[8] Ginestra Bianconi, Albert-Laszlo Barabasi. Competition and multiscaling in evolving networks[J]. Europhysics Letters, 2001, 54(4): 436-442.

[9] Joseph S. Kong, VP Roychowdhury. Preferential survival in models of complex ad hoc networks[J]. Physica A, 2008, 387 : 3335–3347.

[10] Chung F, Lu L. Coupling online and offline analyses for random power law graphs[J]. Internet Mathematics, 2004, 1 (4): 409–461.

[11] Cooper C, Frieze A, Vera J. Random deletion in a scale-free random graph process[J]. Internet Mathematics, 2004, 1 (4): 463–483.

[12] SH Yook, H Jeong, and AL Barabási. Weighted Evolving Networks[J]. PHYSICAL REVIEW LETTERS, 2001, 86(25): 5835-5838.

[13] Dafang Zheng, Steffen Trimper, Bo Zheng, et al. Weighted scale-free networks with stochastic weight assignments[J]. PHYSICAL REVIEW E, 2003, 67: 040102(1-4).

[14] Nitin Arora, Narayanan B, Pau l S. Financial influences and scale-freenetworks. Lecture Notes in Computer Science[J], 2006: 16-23.

[15] ]Weiqiang Huang, Xintian Zhuang, Shuang Yao. The analysis of topological properties and clustering structure of Chinese stock associated network [J]. Management Science, 2008, 21 (3): 94 103.(in Chinese)

[16] Alain Barrat, Marc Barthelemy, Alessandro Vespignanil. Weighted Evolving Networks: Coupling Topology and Weight Dynamics[J]. PHYSICAL REVIEW LETTERS, 2004, 92 (22): 228701 (1-4).

[17] Albert-Laszlo Barabasi, Reka Albert. Emergence of Scaling in Random Networks[J]. Science, 1999, 286: 509-512.

[18] Joseph S. Kong, Nima Sarshar, VP Roychowdhury. Experience versus talent shapes the structure of the web[J]. PNAS, 2008,105(37): 13724-13729.

[19] Sarshar N, Roychowdhury V. Scale-free and stable structures in complex ad hoc networks[J]. Physical Review E, 2004, 69 (2): 357-368