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Physica A 384 (2007) 725–732

www.elsevier.com/locate/physa

A new community-based evolving network model

Zhou Xie, Xiang Li�, Xiaofan Wang

Lab of Complex Networks and Control, Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, PR China

Received 24 March 2006; received in revised form 11 February 2007

Available online 18 May 2007

Abstract

In order to describe the community structure upon the dynamical evolution of complex networks, we propose a new

community based evolving network (CBEN) model having increasing communities with preferential mechanisms of both

community sizes and node degrees, whose cumulative distribution and raw distribution follow scale-invariant power-law

distributions P(SXs)�s�n and P(k)�k�g with exponents of nX1 and gA[2, +N), respectively. Besides, complex networks

generated by the CBEN model are hierarchically structured, which cover the range from disassortative networks to

assortative networks.

r 2007 Elsevier B.V. All rights reserved.

Keywords: Complex network; Community; Preferential growth; Scale free; Hierarchy

1. Introduction

In recent years, the study of complex networked systems [1–4], including the Internet, the World Wide Web(WWW), social networks, biological networks, etc., has been an attractive issue. Historically, the mostdistinctive statistical properties that networks seem to share led to a number of important progresses in ourunderstanding of complex networks, such as the ubiquity of small-world pattern [5], and scale-invariantfeature of power-law degree distribution P(k)�k�g [6]. Another very important characteristic owned in manyreal networks is the presence of communities, which is argued to be the signature of the hierarchical nature incomplex networks [7–9].

Generally speaking, communities are defined as subsets of nodes within which connections are denser whilebetween which are much sparser [10]. Therefore, finding communities in a network plays a crucial role inunderstanding the internal structure and principle function of clusters, which has been the subject ofdiscussions in various disciplines. Although the problem of community detection is very difficult, researchershave proposed some very effective algorithms to find communities in networks having complex topologies[11–15].

To study the effects of community structure on network properties and dynamics, a natural question is:‘‘How can we construct an evolving network model in the community structure?’’ With efforts of manyresearchers, several models for communities in social and biological networks have been proposed [16,17].

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

ysa.2007.05.031

ing author.

ess: xli@sjtu.edu.cn (X. Li).

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The networked seceder model was proposed to construct a community structured network that emerged as aneffect of the agents personal rationales [16]. An evolving model with merging building blocks was proposed tomimic the community structure in social networks and modules in biological systems [17]. As the realization ofcommunity structure in the World Trade Web, it was found that the preferential attachment mechanism is alsowithin the local economic cooperative regions [18,19], i.e., the so-called local world phenomenon [20].Recently, an evolving network model with inner-community preferential attachment, which owns thepower-law degree distribution of the fixed exponent 3 [21], was built.

It has been pointed out that, like degree distributions, the size distribution of communities follows thepower-law scaling property [11–15]. Especially, it has been pointed out that, like node degrees, the preferentialmechanism also exists in the growing evolution of communities. That is, when establishing new links betweencommunities or adding a new node to an existing community, communities with larger sizes are selected withhigher probabilities [22]. Therefore, to demonstrate this phenomenon of ‘‘richer get richer’’ in communitiesscale, we develop the mechanism of community size preferential attachment in our newly proposed evolvingnetwork model with increasing communities in this paper.

The rest of this paper is organized as follows: In Section 2, we propose a new community based evolvingnetwork (CBEN) model. The growing dynamics of the CBEN model, including the community size and degreedistributions, is analyzed in Section 3, where their scaling exponents are analytically studied. After theverification of numerical experiments in Section 4, the whole paper is finally concluded in Section 5.

2. Model description

In our CBEN model, an undirected and unweighted network is initialized with c0(c041) communities, eachof which has n0 fully connected nodes. There are c0(c0�1)/2 inter-community links to make c0 communitiesfully connected. In each community, the node to which inter-links connect is selected at random. The model isevolved with the following scheme.

2.1. Growth

At each time step, a new community containing n0 fully connected nodes is added with probability p, and werandomly choose one node in the new community to connect m nodes in other existing communities followingthe preferential attachment mechanisms.

A new node is added with probability 1�p at each time step. First, it chooses which community to add into,and then it connects m already existing nodes in the network. Each one of those m nodes is chosen from itsown community with probability q and from other communities with probability 1�q. Here q is assumed to beclose to 1, or at least larger than 0.5, to guarantee that the connections of nodes within communities are denserthan those between them.

Please note that when a new node chooses an inter-community neighbor, it firstly chooses a community andthen chooses one node from that community. Although one certain community could be chosen for more thanone time, multiple connections to one and the same node are not permitted.

2.2. Preferential attachment rules

(a)

Community size preferential attachment: When a new node chooses an existing community to add (orchooses another community from which to get an inter-community neighbor), we assume the probabilityP(Si) that it will choose community i depends on the size Si of community i, such thatY

ðSiÞ ¼SiPkSk

. (1)

(b)

Degree preferential attachment: When choosing a neighbor, a new node firstly chooses a community i,which maybe its own community or another community according to Eq. (1), and then connects with onenode in it. We assume the probability P(Kij) that the new node will connect to node j in community i is

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Fig. 1. An illustration of the CBEN model. (a) The initial network with three communities, each of which has four fully connected nodes.

(b) A new node joins an existing community according to Eq. (1). It has three links. With probability q, it chooses two inner-community

neighbors according to Eq. (2). With probability 1�q, it chooses an inter-community neighbor by firstly choosing another community

according to Eq. (1) and then a node in that community according to Eq. (2). (c) Depending on probability p, a new community with four

fully connected nodes is created, which connects to three nodes in other communities. It firstly chooses other communities according to (1),

and then chooses one node in each of them according to (2).

Z. Xie et al. / Physica A 384 (2007) 725–732 727

described byYðKijÞ ¼

Kij þ aPkðKik þ aÞ

, (2)

where Kij stands for the degree of node j in community i and aA[�m, +N).In the CBEN model, the probabilities should satisfy 0ppp1 and 0pqp1. When p ¼ 0 (or p ¼ 1), there is

no formation (or always formations) of new communities in the evolution of network. Fig. 1 is the schematicdiagram of the CBEN model.

3. Model analysis

Using the mean-field method [23], we analytically arrive at the distribution of the community sizes and nodedegrees with exact scaling exponents.

3.1. Power-law distribution of community sizes

Assume the sizes of communities are continuous. Therefore, the size preferential probability (1) can beinterpreted as a continuous rate of change of Si. Consequently, for community i, we have

qSi

qt¼

SiPkSk

. (3)

Notice thatP

kSk ¼ N, which leads to

qSi

qt¼

SiPkSk

¼Si

N¼

Si

c0n0 þ ð1� pþ pn0Þt�

Si

ð1� pþ pn0Þt, (4)

whose solution, with the initial condition that community i was added to the network at time ti with sizeSi ¼ n0, is

SiðtÞ ¼ n0ðt=tiÞ1=1�pþpn0 . (5)

Then the probability that a community has a size Si(t) not less than s, P(Si(t)Xs), is

PðSiðtÞXsÞ ¼ P tipn1�pþpn00

s1�pþpn0t

!. (6)

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For simplicity, we assume that the communities were added at equal time intervals to the system, thus theprobability density of ti is

PðtiÞ ¼1

c0 þ pt. (7)

Substituting Eq. (7) into Eq. (6), we obtain

PðSiðtÞXsÞ ¼ P tipn1�pþpn00

s1�pþpn0t

!¼

n1�pþpn00

s1�pþpn0

1

c0 þ pt(8)

which implies that the cumulative distribution of community size obeys a power-law P(SXs)�s�n with theexponent n ¼ 1+p(n0�1)X1. Because many real networks have been reported [12–15] to have their exponentsnA[1,2], they fall into the CBEN model with the parameter requirement of pA[0, (n0�1)

�1].

3.2. Power-law distribution of node degrees

In order to analyze the degree distribution, we divide the degree of each node into two parts: the inner-community degree, which is the number of links connecting to other nodes within the same community,and the inter-community degree, which means the number of links connecting to nodes in other communities.We firstly derive the inner-community degree distribution as follows.

When a new community is created with probability p, the inner-community degrees of nodes in the alreadyexisting communities will not change over time. On the other hand, when a new node is created withprobability 1�p, it firstly chooses a community i to add into according to Eq. (1). Then with probability q, itchooses inner-community neighbors according to Eq. (2). Consequently, for node j in community i

qKijðinnerÞ

qt¼ ð1� pÞqm

YðSiÞ

YðKijÞ

¼ ð1� pÞqmSiPkSk

Kij þ aPkðKik þ aÞ

. ð9Þ

Independent of the initial condition, we haveP

kSk ¼ N, andP

k(Kik+a) ¼ [(1+q)m+a]Si. Therefore

qKijðinnerÞ

qt¼ ð1� pÞqm

Si

N

Kij þ a½ð1þ qÞmþ a�Si

¼qmð1� pÞðKij þ aÞ

½ð1þ qÞmþ a�ð1� pþ pn0Þt¼ a

Kij

tþ b

1

t, ð10Þ

where

a ¼qmð1� pÞ

½ð1þ qÞmþ a�ð1� pþ pn0Þ,

b ¼aqmð1� pÞ

½ð1þ qÞmþ a�ð1� pþ pn0Þ.

The solution of Eq. (10), with the initial condition that node j was added to community i at time tj with qm

inner-community connections, is

KijðinnerÞðtÞ ¼ �b

aþ qmþ

b

a

� �t

ti

� �a

. (11)

Assume that we add a community or a node to the system at equal time intervals. Then the pro-bability density of ti P(ti) ¼ 1/t. Therefore, the probability that a node has an inner-community degree

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smaller than k is given by

PðKijðinnerÞðtÞokÞ ¼ P ti4qmþ b=a

k þ b=a

� �1=a

t

!

¼ 1�qmþ b=a

k þ b=a

� �1=a

. ð12Þ

Then the probability density for P(Kij(inner)) can be obtained using

PðKijðinnerÞðtÞÞ ¼qPðKijðinnerÞðtÞokÞ

qk¼

1

a

ðqmþ b=aÞ1=a

ðk þ b=aÞ1þ1=a. (13)

The inter-community connections are generated from two means. On the one hand, when a community isadded with probability p, it creates m inter-community connections and on the other hand, the m connectionsof a new node are introduced as inter-community connections with probability 1�q. Although the inter-community degrees of nodes are very difficult to analyze, in most situations, the birth of new communities aremuch rarer than the creation of new nodes, and connections between communities are much sparser thanthose within communities [10]. As a result, we have the degree of node j in community i

Kij ¼ KijðinnerÞ þ KijðinterÞ � KijðinnerÞ, (14)

and the probability density for P(Kij) is

PðKijÞ � PðKijðinnerÞÞ ¼1

a

ðqmþ b=aÞ1=a

ðk þ b=aÞ1þ1=a, (15)

which implies that the CBEN model owns a power-law degree distribution of P(k)�k�g with an alterableexponent g ¼ 1+1/a. Since p, qA[0,1] and aA[�m, +N), the valid range of a falls into [0,1], and gA[2, +N).Therefore, combined with the investigation of community size distribution in the previous section, the CBENmodel captures the distributions of node degrees and community sizes of real-life complex networks.

4. Numerical simulations

We initialize a network having three communities, any of which has 4 nodes fully connected, i.e., c0 ¼ 3 andn0 ¼ 4. The network evolve with the model parameters of p ¼ 0.02, q ¼ 0.9, a ¼ �1 and m ¼ 3, which ownsthe scale-free distributions of community sizes and node degrees with the theoretically predicted exponentsv ¼ 1.06 and g ¼ 2.8, respectively. After T ¼ 50,000 time steps, we have a network with 53,192 nodes in 1063communities, whose cumulative distribution of communities sizes is shown in Fig. 2, obeying a power law withan exponent n ¼ 1. In Fig. 3, we plot the degree distributions of the network, including the inner-communitydegree and total degree, which follow a power-law distribution with the exponent g ¼ 2.7. It is clearly observedthat both actual exponents are very close to the theoretical predictions. Fig. 3 also shows that the inner-community degrees and total degrees of nodes are almost the same, owing to the connections betweencommunities are much sparser than those within communities.

After calculations, the above-constructed CBEN model has diameter D ¼ 10, the average path lengthl ¼ 5.408, and the average clustering coefficient C ¼ 0.1088, which, compared with the same-scale randomnetworks, captures the small-world pattern of small average path length with large clustering. In Fig. 4, thedegree-clustering correlation (clustering spectra) is plotted for a network generated with the above same initialconditions for T ¼ 10,000 time steps, and the scaling law C(k)�k�b with b ¼ 0.75, which indicates complexnetworks generated by the CBEN model is hierarchical [24]. Please note that the hierarchy we mentioned hereis in the sense of the network structure itself, not the community structure in complex networks.

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Fig. 2. Cumulative distribution of the community sizes in the CBEN model. The gradient of the straight line is �1.

Fig. 3. The degree distributions of nodes in the CBEN model. The gradient of the straight line is �2.7.

Z. Xie et al. / Physica A 384 (2007) 725–732730

To anatomize the network hierarchy, an assortative mixing coefficient r is as is the following [25]:

r ¼M�1

Pijiki � M�1

Pi12ðji þ kiÞ

� �2M�1

Pi12ðj2i þ k2

i Þ � M�112

Piðji þ kiÞ

� �2 , (16)

where ji, ki are the degrees of the nodes at the ends of the ith edge, with i ¼ 1, 2,y, M and M is the number oflinks in the network. Generally, social networks are found to be assortative with positive r, and many

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Fig. 4. The clustering spectra C(k) obtained in the CBEN model by averaging the results of 50 runs.

Fig. 5. The variation of r as p increases in the CBEN model by averaging the results of 50 runs.

Z. Xie et al. / Physica A 384 (2007) 725–732 731

technological networks are disassortative with negative r [25]. Because the community structure is popular inboth social and technological networks, we observe the influence of p on r in our CBEN model, under thesame other initial conditions with the evolution of T ¼ 10,000 time steps. As shown in Fig. 5, the assortativemixing coefficient r increases when p increases from 0.01 to 1. In particular, when po0.08, the generatednetworks are disassortative with ro0. When 0.08opp1, the generated networks are assortative with r40.Therefore, the CBEN model covers the range from disassortative networks to assortative networks.

5. Conclusions

In this paper, we have proposed a new community-based evolving network model with the preferenceof both community sizes and node degrees, which yields scale-free networks having power-law distributionsof community sizes and node degrees with arbitrary scaling exponents of nX1 and gA[2, +N), respectively.

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The small-world pattern and hierarchy property have been numerically investigated in this new model, whichis capable of generating complex networks from the social to technological fields.

Acknowledgments

This work was partly supported by the National Natural Science Foundation of P. R. China under Grantnos. 70431002 and 60504019, and the NSFC/KOSEF program under Grant no. 60611140548, and the 973Fund under Grant no. 2006CB708302.

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