ARTICLE IN PRESS

0378-4371/$ - se

doi:10.1016/j.ph

$This resear

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Physica A 381 (2007) 515–524

www.elsevier.com/locate/physa

Clustering coefficients of growing networks$

Dinghua Shia,b, Stuart X. Zhuc, Liming Liud,�

aDepartment of Mathematics, Shanghai University, Shanghai 200444, ChinabCenter for the Study of Complex Networks, Shanghai Academy of Systems Science, China

cDepartment of Information and Systems Management, Hong Kong University of Science and Technology, Hong Kong, ChinadDepartment of Logistics, Hong Kong Polytechnic University, Hong Kong, China

Received 14 November 2005; received in revised form 29 March 2007

Available online 13 April 2007

Abstract

In this paper, we develop a general analytical method to compute clustering coefficients of growing networks. This

method can be applied to any network as long as we can construct and solve the dynamic equation for the degree of any

node. We also verify the accuracy of the method through simulation.

r 2007 Elsevier B.V. All rights reserved.

Keywords: Growing network; Clustering coefficient; Dynamic equation

1. Introduction

Clustering is an important network concept originated in sociology. In a social network, clusteringcoefficient measures the extent to which the members of a circle of friends and acquaintances are mutualfriends or acquaintances. Such a measure is obviously also of great interest for other complex networks, and itis useful to obtain analytical clustering coefficients in growing networks. There are two approaches in existingliterature on defining and computing the clustering coefficients: through the actual number of links in theneighborhood of a node or through the number of triangles in the network.

1.1. Type-1 clustering coefficients

Watts and Strogatz [1] provide a simulation analysis of the clustering coefficient of a small-world network.The clustering coefficient of an arbitrary node l is defined by the authors as the ratio of the actual number oflinks EðlÞ and the possible number of links among all the nodes in the neighborhood of node l

ClðNÞ ¼2EðlÞ

klðNÞ½klðNÞ � 1�, (1)

e front matter r 2007 Elsevier B.V. All rights reserved.

ysa.2007.03.054

ch is supported in part by the National Natural Science Foundation of China through Grant 70171059, by the 863 Project

2002AA234021 and by Hong Kong Research Grant Council through Grants PolyU6189/01E and PolyU6133/02E.

ing author. Tel.: +86852 2766 5009; fax: +86852 2330 2704.

ess: lgtliulm@polyu.edu.hk (L. Liu).

ARTICLE IN PRESSD. Shi et al. / Physica A 381 (2007) 515–524516

where klðNÞ is the degree of note l and N the size of the network. The network clustering coefficient is then theaverage of all ClðNÞ

Cð1ÞðNÞ ¼

PNi¼1CiðNÞ

N. (2)

Referred to as the type-1 clustering coefficient, we note from (1) that the key to computing Cð1ÞðNÞ is EðlÞ.Barrat and Weigt [2] give the following analytical expression for small-world networks:

Cð1ÞðNÞ ¼3ðk � 2Þ

4ðk � 1Þð1� pÞ3,

where k is the degree of each node in the original regular network and p is the probability of rewiring a link.Albert and Barabasi [3] find, through simulation, that the clustering coefficient of a scale-free BA network

decreases in the network size and they approximate it by Cð1ÞðNÞ�N�0:75.Klemm and Eguıluz [4] construct a model to generate growing networks of varying clustering levels, with

the scale-free BA network at one extreme and a highly clustered small-world network at the other extreme.For the BA network, the authors give

Cð1ÞðNÞ ¼

RN

1dipðliÞ

RN

1djpðljÞpðijÞ

klðNÞ2

¼ mðlnNÞ2=ð8NÞ,

where k2l ðNÞ is an approximation of klðNÞ½klðNÞ � 1� and pðijÞ is the probability that there exists a link from

node j to node i ðiojÞ.By the mean-field approach, Fronczak et al. [5] provide a more accurate expression for the BA network

Cð1ÞðNÞ ¼m2ðmþ 1Þ2

4ðm� 1Þln

mþ 1

m�

1

mþ 1

� �ðlnNÞ2

N.

In studying correlated random networks with hidden variables, Bogu ~na and Pastor-Satorras [6] alsoconsider the network clustering coefficient and provide an explicit expression when a specific distribution ofhidden variable and a symmetric connection probability function are given.

1.2. Type-2 clustering coefficients

Newman et al. [7] define the following ratio as the network clustering coefficient:

Cð2ÞðNÞ ¼3� number of triangles in the network

number of connected triples of vertices. (3)

For bipartite graphs, such as an actor co-appearance network, the authors simplify (3) to an explicitapproximation

1

Cð2ÞðNÞ� 1 ¼

ðm2 � m1Þðn2 � n1Þ2

m1n1ð2n1 � 2n2 þ n3Þ,

where mn and nn are the nth moments of the distribution of the number of movies in which the actors haveappeared and the number of actors in a movie, respectively.

Clearly, the key to computing the type-2 clustering coefficient (3) of a network is the number of triangles inthe network.

In discussing why social networks are different from other types of networks, Newman and Park [8] usegenerating functions to derive an asymptotic expression, Cð2ÞðNÞ�N ð7�3gÞ=ðg�1Þ, for the clustering coefficient ofa general random network with a degree exponent g.

Independent of Ref. [7], but without considering clustering coefficients, Gleiss et al. [9] give the first generalexpression for the number of triangles in growing networks

hDi ¼X

rosot

prspstprt, (4)

ARTICLE IN PRESSD. Shi et al. / Physica A 381 (2007) 515–524 517

where pij is the probability that there exists a link from node j to node i ðiojÞ. For the BA model, the authorsreduce (4) to hDi�ðlnNÞ3 for b ¼ 1

2 and hDi�N2b�1 lnN for 12obo1.

Bianconi and Capocci [10] consider the number of size h loops in the BA network, and obtain the sameresult as in Ref. [9] for the number of triangles by letting h ¼ 3.

Bollobas and Riordan [11] define the clustering coefficient of a growing network as the weighted average ofthe clustering coefficients of all nodes,

Cð3ÞðNÞ ¼XN

l¼1

EðlÞ

,XN

l¼1

klðNÞ

2

� �, (5)

where the weights are proportional to the number of nodes in the neighborhood. The authors note that Cð3Þ isequivalent to Cð2Þ, since the denominator in (5) represents the number of connected triples of all the vertices inthe network while the numerator in (5) represents the corresponding number of connected triples of thetriangles. For a BA network, they show that

Cð3ÞðNÞ ¼ ðm� 1ÞðlnNÞ2=ð8NÞ as N !1.

In summary, analytical results for network clustering coefficients are still limited. For scale-free net-works, the existing literature is mainly on the original BA network, i.e., Refs. [4,5,10,11]. While the formulain Ref. [9] for the number of triangles is quite general, it cannot be used to compute Cð1ÞðNÞ. Like Ref. [4],the formula given in Ref. [6] requires symmetric connection probability functions and is thus applicableonly to the basic BA type of networks. For more details on the existing literature on clustering in complexnetworks, one may refer to Refs. [3] and [12]. Degree-dependent clustering coefficients have been studiedby a number of authors. The literature, see, for example, Refs. [4,13], is mainly on high clustering networkswhere the clustering coefficients are usually independent of the network size which are out of the scope of thispaper.

In this paper, we develop a general method to compute clustering coefficients using EðlÞ. By Eqs. (1), (2) and(5), our method can be used to compute Cð1ÞðNÞ, Cð2ÞðNÞ and Cð3ÞðNÞ from EðlÞ for a wide range of growingnetworks. The remainder of this paper is organized as follows. In Section 2, we present a general approach forcomputing the clustering coefficients of growing networks from the dynamic equations, and then provideanalytical formulas for the clustering coefficients of stationary growing networks, nonstationary growingnetworks, and growing random networks. We show that by computing EðlÞ, we can also compute hDi andhence Cð2Þ. In Section 3, we verify our method and results through simulation.

2. Clustering coefficients in growing networks

2.1. The basic approach

Consider node i, added at time step i, in a growing network. By the continuum theory, its degree kiðtÞ attime step t generally satisfies the following dynamic equation:

qkiðtÞ

qt¼ f ðkiðtÞÞ. (6)

Let mðiÞ be the number of links added with node i. Integrating (6), we have

kiðtÞ ¼ mðiÞ þ

Z t

i

f ðkiðjÞÞdj. (7)

Two observations can be drawn from Eq. (7): the expected number of links connecting node i with the nodesadded from time step i þ 1 to time step t is given by

R t

if ðkiðjÞÞdj; and the probability pðijÞ that there exists a

link from node j to node i (j4i) equals f ðkiðjÞÞ.For a growing network, we cannot count the actual number of connections. Instead, we use the expected

number of connections to represent EðlÞ, and we need to consider the order by which node l and its neighborsappear. For example, when the size of the network reaches N and loioj (node l joins the network first andnode j joins the last), the expected number of links between l’s neighbors i and j is given by

ARTICLE IN PRESSD. Shi et al. / Physica A 381 (2007) 515–524518

RN

ldipðliÞ

RN

idjpðljÞpðijÞ. Similarly, we can compute the expected numbers of links for five other cases: ioloj,

iojol, lojoi, joloi and joiol, and the expected total number of links among all the neighbors of l isgiven by

EðlÞ ¼1

2

Z N

l

dipðliÞ

Z N

i

djpðljÞpðijÞ þ

Z l

1

dipðilÞ

Z N

l

djpðljÞpðijÞ

�

þ

Z l

1

dipðilÞ

Z l

i

djpðjlÞpðijÞ þ

Z N

l

dipðliÞ

Z i

l

djpðljÞpðjiÞ

þ

Z N

l

dipðliÞ

Z l

1

djpðjlÞpðjiÞ þ

Z l

1

dipðilÞ

Z i

1

djpðjlÞpðjiÞ

�,

where since the links between any two nodes are counted twice, we divide the sum of the six integrations by 2.Changing the order of integrations, we obtain

EðlÞ ¼

Z N

l

dipðliÞ

Z N

i

djpðljÞpðijÞ þ

Z l

1

dipðilÞ

Z N

l

djpðljÞpðijÞ

þ

Z l

1

dipðilÞ

Z l

i

djpðjlÞpðijÞ. ð8Þ

The key step of our approach is to obtain Eq. (8) explicitly, with which we then have

ClðNÞ ¼ 2EðlÞ=k2l ðNÞ, (9)

where in the denominator, k2l ðNÞ is an approximation of klðNÞ½klðNÞ � 1�. By (2), the average clustering

coefficient of a growing network is given by

Cð1ÞðNÞ ¼

RN

1 2EðlÞ=k2l ðNÞdl

N. (10)

2.2. Stationary growing networks

For many stationary growing networks, the degree of a node as a function of the network size and theprobability pðijÞ that a link exists between i and j can be approximated, respectively, by

klðNÞ�ðN=lÞb and pðijÞ�jb�1

ib, (11)

where b (0obo1) is the network dynamic exponent. When these two relations are satisfied, we can computethe three integrations in (8) as follows. For loioj,Z N

l

dipðliÞ

Z N

i

djpðljÞpðijÞ�

Z N

l

ib�1

lbdi

Z N

i

jb�1

lb�

jb�1

ibdj

¼ l�2bN2b�1ðlnN � ln lÞ

2b� 1�

N2b�1 � l2b�1

ð2b� 1Þ2

" #.

For ioloj,Z l

1

dipðilÞ

Z N

l

djpðljÞpðijÞ�

Z l

1

lb�1

ibdi

Z N

l

jb�1

lb�

jb�1

ibdj

¼ l�1ð1� l�2bþ1ÞN2b�1 � l2b�1 þ 1

ð2b� 1Þ2.

ARTICLE IN PRESSD. Shi et al. / Physica A 381 (2007) 515–524 519

For iojol,Z l

1

dipðilÞ

Z l

i

djpðjlÞpðijÞ�

Z l

1

ðl=iÞb

ldi

Z l

i

ðl=jÞb

l�ðj=iÞb

jdj

¼ l2b�2ln l

2b� 1þ

l�2bþ1 � 1

ð2b� 1Þ2

" #.

Combining the right-hand sides of the above relations, we obtain the expected number of connectionsamong the neighbors of node l,

EðlÞ�N2b�1½l�2bð2b� 1ÞðlnN � ln lÞ � 2l�2b þ l�1�

ð2b� 1Þ2þ

l2b�2ð2b� 1Þ ln l þ 3l�1 � 2l2b�2

ð2b� 1Þ2

( ). (12)

Substituting EðlÞ into (9) and (10), and noting that klðNÞ�ðN=lÞb, the clustering coefficient of an arbitrarynode l and the network clustering coefficient are given, respectively, by

ClðNÞ�½ð2b� 1ÞðlnN � ln lÞ � 2þ l2b�1�

ð2b� 1Þ2Nþ

l4b�2ð2b� 1Þ ln l þ 3l2b�1 � 2l4b�2

ð2b� 1Þ2N2b

( )(13)

and

Cð1ÞðNÞ ¼

Z N

1

ClðNÞdl=N

�N�2b�1

ð2b� 1Þ2½2bþ 3ð2bÞ�1 � 3�N2b þ

2b� 1

4b� 1N4b�1 lnN

�

þ1

2b�

10b� 3

ð4b� 1Þ2

� �N4b�1 � ½ð2b� 1ÞðlnN þ 1Þ þ ð2bÞ�1 � 2�N2b�1 þ

10b� 3

ð4b� 1Þ2�

3

2b

�. ð14Þ

Expression (14) can be simplified according to the value of b by keeping only the appropriate terms withhigher orders of N on its right-hand side. For 0obo1

2, keeping the first two terms, we have

Cð1ÞðNÞ�ðN�1 þN2b�2 lnNÞ; 0obo1

2. (15)

We point out that if we only keep the first term (which has the highest order of N when N is sufficiently large),the result will not be sufficiently accurate, as demonstrated later in Fig. 2.

Substituting b ¼ 12and applying the L’Hospitale law, (14) is reduced to

Cð1ÞðNÞ�ðlnNÞ2=N; b ¼ 12. (16)

For 12obo1, we keep only the second term and

Cð1ÞðNÞ�N2b�2 lnN; 12obo1. (17)

With Eqs. (15)–(17), we can obtain the asymptotic network clustering coefficient of any stationary growingnetwork satisfying (11), as shown by the following three typical networks.

(1) The BA network: The dynamic equation for this network is qkiðtÞ=qt ¼ kiðtÞ=ð2tÞ, with solutionklðtÞ ¼ mðt=lÞ1=2. Thus, b ¼ 1

2and the second expression in (16) gives the network clustering coefficient. Noting

that pðijÞ ¼ pðjiÞ ¼ ðm=2ÞðijÞ�1=2 and is symmetric, the result can also be obtained from (9) directly,

ClðNÞ ¼

RN

1 dipðliÞRN

1 djpðljÞpðijÞ

klðNÞ2

�ðlnNÞ2

N. (18)

Noting that since ClðNÞ is independent of l, (18) also gives the network clustering coefficient Cð1ÞðNÞ.Klemm and Egu�{luz [4] obtained (18), but their result cannot be extended to other models.

ARTICLE IN PRESSD. Shi et al. / Physica A 381 (2007) 515–524520

(2) The modified BA network with linear preferential attachment [14]: From the dynamic equation

qkiðtÞ

qt¼

m½ð1� pÞki þ p�

½2mð1� pÞ þ p�t,

we have

klðtÞ ¼ mþp

1� p

� �t

l

� b�

p

1� p,

where p is the probability that a new node is randomly connected to the existing nodes. Thus,b ¼ mð1� pÞ=½2mð1� pÞ þ p�. Noting that, 0obp1

2for 0ppo1, the clustering coefficient is given by (15)

or (16).(3) The initial attractiveness network [15]: Dorogovtsev and Mends [15] use the master equation method to

study the network degree problem. They find

klðtÞ ¼ At

l

� m=ðmþAÞ

� A, (19)

where A ðA40Þ is the initial attractiveness. Noting that k ¼ qþ A where q is the (average) number of theincoming links of a node, we have the corresponding dynamic equation qq=qt ¼ mq=ðmþ AÞt, and henceb ¼ m=ðmþ AÞ. Here, for 0oA, 0obo1 and the corresponding clustering coefficients are given by (15)–(17),respectively.

2.3. Nonstationary growing networks

Our method can also be applied to some nonstationary growing networks. Consider the power-growing BAnetwork proposed in Ref. [16]. The dynamic equation is qkiðtÞ=qt ¼ ð1þ yÞkiðtÞ=2t, where 0pyo1. Under theinitial condition klðlÞ ¼ mly, Shi et al. [17] obtain klðtÞ ¼ mtyðt=lÞb, where b ¼ ð1� yÞ=2. Noting thatpðijÞ ¼ ðm=2Þð1þ yÞðijÞ�b, and by the symmetry of pðijÞ, (9) can be simplified to

ClðNÞ ¼

RN

1 dipðliÞRN

1 djpðljÞpðijÞ

klðNÞ2

�ðNy � 1Þ2

N1þy . (20)

Since ClðNÞ is independent of l, Cð1ÞðNÞ equals ClðNÞ.Shi et al. [17] propose a logarithmic-growing BA network with growth function m ln t. Separately,

Krapivsky and Rendner [18] show the validity of such a logarithm model by real network examples andidentify the mechanism by which the links of a network grow logarithmically. Since the solution to thedynamic equation is not available for this nonstationary network, our method cannot be applied, and we needto develop a different method for the clustering coefficient, perhaps a numerical method similar to the Markovchain method in Ref. [17] for the degree distribution.

2.4. Growing random networks

Now, we show that our method can also be applied to growing random networks. Let us revisit the modelproposed in Ref. [14]. When p ¼ 1, the modified BA model represents a growing random network with b ¼ 0.In this case, pðijÞ ¼ m=j: By (8), we haveZ N

l

dipðliÞ

Z N

i

djpðljÞpðijÞ� l�1 �N�1 �lnN � ln l

N

� �,

Z l

1

dipðilÞ

Z N

l

djpðljÞpðijÞ�l � 1

lðl�1 �N�1Þ,

Z l

1

dipðilÞ

Z l

i

djpðjlÞpðijÞ�l � 1� ln l

l2.

ARTICLE IN PRESSD. Shi et al. / Physica A 381 (2007) 515–524 521

Using klðNÞ ¼ ðlnN � ln l þ 1Þm instead of klðNÞ � m lnN as in Ref. [14] and keeping the highest order term,we obtain ClðNÞ� 3l�1=ðlnN � ln l þ 1Þ2: It is easy to see that

Cð1ÞðNÞ ¼

Z N

1

ClðNÞdl=N�1

N�

1

NðlnN þ 1Þ: ð21Þ

2.5. Computing the number of triangles

We note that hDi ¼ 1=3RN

1 EðlÞdl. Thus, as long as we can compute EðlÞ, we can also compute Cð2ÞðNÞ. Inparticular, for stationary growing networks, we can compute hDi from (12).

For example, we can easily compute hDi in the BA network. For b ¼ 12, by the L’Hospitale law, we obtain

EðlÞ� ðlnNÞ2=l. Then, hDi� ðlnNÞ3; For 12obo1, we only keep the first term of EðlÞ that has the highest order

of N, i.e., EðlÞ�N2b�1=l. Then, hDi�N2b�1 lnN; For the case when 0obo12, we have EðlÞ� 1=l by keeping

the highest order of N, and hence hDi� lnN.The results for the cases with 1

2pbo1 are identical to those given in Ref. [9].

3. Simulation comparisons

We demonstrate the effectiveness of the method by comparing the analytical results with the correspondingsimulation results numerically for five cases.

(1) The BA model with m ¼ 4 and b ¼ 12: To verify our simulation, we use two different network models to

generate clustering coefficient curves. Specifically, when m ¼ 4, both the initial attractiveness model withA ¼ 4 and the modified BA model with p ¼ 0 reduce to the same BA model, so that the two clusteringcoefficient curves from simulation should overlap. This is indeed the case as shown in Fig. 1. Fig. 1 also showsthat the analytical clustering coefficient for the BA model by (16) is very close to the simulation results whilethe clustering coefficient from the approximation formula in Ref. [3] is inaccurate for small and large N.

(2) The initial attractiveness model and the modified BA model: We consider an initial attractiveness modelwith m ¼ 4 and A ¼ 7 and a modified BA model with m ¼ 4 and p ¼ 3

4; both with b ¼ 4

11. We also verify our

simulation with two different models mentioned above. As shown in Fig. 2, our analytical result by (15), i.e.,

102 103 104 1053

4

5

6

7

8

9

10

11

N

(NC)1/2

Fig. 1. The clustering coefficients from the analytical formula (15) with b ¼ 12in solid red line; from the approximation formula

Cð1ÞðNÞ�N�0:75 in blue dash line; and from simulation in black circles for the initial attractiveness model with m ¼ 4 and A ¼ 4 and in

black triangles for the modified BA model with m ¼ 4 and p ¼ 0.

ARTICLE IN PRESS

102 103 104 105

104

103

102

101

100

N

C

Fig. 2. The clustering coefficients from the analytical formula (16) with b ¼ 411

in solid red line, from the approximation formula

Cð1ÞðNÞ� 1=N in dash blue line, and from simulation in black circles for the initial attractiveness model with m ¼ 4 and A ¼ 7 and in black

triangles for the modified BA model with m ¼ 4 and p ¼ 34.

102 103 104 1055

10

15

20

N

N2/3C

Fig. 3. The clustering coefficients from the analytical formula (17) with b ¼ 23in solid red line and from simulation in black circles for the

initial attractiveness model with m ¼ 4 and A ¼ 2.

D. Shi et al. / Physica A 381 (2007) 515–524522

the solid red line, matches the simulation very closely. However, if we only use 1=N, the highest order term ofN, the result is not sufficiently accurate as shown by the dash blue line.

(3) The initial attractiveness model with m ¼ 4, A ¼ 2 and b ¼ 23: For the initial attractiveness model, Fig. 3

shows that the analytical result from (17) matches the simulation result closely. Because of the property of thefunction ln x=xa where a40, we observe that for larger b, the network clustering coefficient is increasing firstand then decreasing with the number of nodes in the network.

(4) The power-growing BA model with m ¼ 4, y ¼ 0:2 and b ¼ 0:4: For the power-growing BA model, (20)is a new analytical result. Fig. 4 shows that the analytical result by (20) matches the simulation result well.

ARTICLE IN PRESS

102 103 104 105104

103

102

101

100

N

C

Fig. 4. The clustering coefficients from the analytical formula (20) with b ¼ 0:4 in solid red line and by simulation in black stars for the

power-growing BA model with m ¼ 4 and y ¼ 0:2.

102 103 104 105

104

103

102

101

100

N

C

Fig. 5. The clustering coefficients from the analytical formula (21) with b ¼ 0 in solid red line, from the approximation formula

Cð1ÞðNÞ� 1=N in dash blue line, and by simulation in black triangles for the modified BA model with m ¼ 4 and p ¼ 1.

D. Shi et al. / Physica A 381 (2007) 515–524 523

(5) The growing random model: The growing random model considered here is equivalent to the modifiedBA model with m ¼ 4, p ¼ 1 and b ¼ 0. Fig. 5 shows that the solid red line computed from (21) overlaps withthe simulation result. The approximation by 1=N in the dash blue line overestimates the clustering coefficientwhen N is relatively small.

In summary, we conclude that the method is reliable and can be used to compute analytical networkclustering coefficients accurately for a variety of network models.

ARTICLE IN PRESSD. Shi et al. / Physica A 381 (2007) 515–524524

4. Conclusion and discussion

We proposed a general method to compute clustering coefficients of growing networks. This method isflexible and can be applied to different network models to find both the clustering coefficient of an arbitrarynode and the clustering coefficient of the network. In fact, except for the clustering coefficient of the originalBA network, i.e., (16), (15) and (17) are new analytical results for clustering coefficients of other stationarygrowing networks while the analytical formulas (20) and (21) are the first clustering coefficient results fornonstationary growing networks and random growing networks, respectively.

For more complicated networks, we need to develop a new method that does not require the explicitsolution to the dynamic equation as noted in Section 2.2. For the computation of the degree-dependentclustering coefficient ðCðkÞÞ of a node with a given degree k, we also need a new approach to compute it forgeneral growing networks.

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