Physica A 391 (2012) 508–514

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Physica A

journal homepage: www.elsevier.com/locate/physa

Synchronization on overlapping community networkJianshe Wu a,∗, Xiaohua Wang b, Licheng Jiao a

a Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education of China, Xidian University, Xi’an 710071, PR Chinab Aeronautical Computing Technique Research Institute, Xi’an 710068, PR China

a r t i c l e i n f o

Article history:Received 18 August 2009Received in revised form 9 August 2011Available online 10 September 2011

Keywords:Complex networkOverlapping communityPhase synchronization

a b s t r a c t

In this paper, we propose a simple random network model with overlapping communitiescontrolled by several parameters, and investigate the influence of the overlappingcommunity structure on the synchronization behavior under different parameters. It isfound that the synchronizability of the network is mainly influenced by the overlappingsize of the communities and the connectivity density of the overlapped group to theother interrelated communities, and has nothing to do with the intra-connectivity of theoverlapped group. In addition, it is found that the highly interconnected communitiescan be almost synchronized in a given time scale, whereas the overlapped group is farfrom synchronization. Furthermore, the instantaneous frequencies of the nodes in thecommunities and their overlapped group are also investigated, which show that the nodesin the overlapped group will exhibit a remarkable oscillation with a weighted meanfrequency of the other correlative communities.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Complex networks are gaining more and more importance as a powerful interpretation tool for many different kindsof natural, biological and social networks [1,2]; a recent introduction to networks has been given by Newman in Ref. [3].A feature of many real-world networks is the community structure [3,4], where the edges inside the community aremuch denser than those connecting the nodes of different communities. Generally, communities are groups of nodeswhich probably share common properties or play similar roles within the network. In addition, the community structurescan dramatically affect the behaviors (dynamical processes or functions) of the complex networks [5–10]. Consequently,investigation of the community structures has important practical significances, and can help one to understand thestructural, functional, and dynamical properties of a network.

Over the past years, a great deal of work has been focusing on the community structure, especially for the disjointcommunity structure; a detailed survey on the problem of community detection has been recently given by Fortunato [11],whereas in many real-world networks communities often overlap to some extent [12,13]. For example, in the network ofresearchers collaboration, an author might work with researchers in two or more groups, in a biological network a proteinmight interact with two or more groups of proteins, and so on. So far, the overlapping community structure has also beenwidely studied [14–22] since it was raised by Palla et al. [14].

However, the vast majority of these investigations are concerned on how to extract the overlapping communities incomplex networks, neglecting its function (or dynamics). In fact, the dynamical behavior (e.g. cooperation of parallel processof the communities) occurring on the network is closely related to the overlapping community structure. Separate studiesabout the overlapping community structure and the dynamical synchronization in a general network (see Refs. [23–25])cannot unravel the relations. Refs. [12,13] show that the dynamics on a network with overlapping community structure is

∗ Corresponding author. Tel.: +86 29 88202279; fax: +86 29 88201023.E-mail addresses: jshwu@mail.xidian.edu.cn (J. Wu), wangxh@mail.xidian.edu.cn (X. Wang), lchjiao@xidian.edu.cn (L. Jiao).

0378-4371/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2011.08.059

J. Wu et al. / Physica A 391 (2012) 508–514 509

a b c

Fig. 1. Example of networks with overlapping community structure, where N = 400, p1 = 0.1, p0 = 0.1 and p11 = 0.001, and the other parametersare: (a) s = 0.1, p01 = 0.1; (b) s = 0.1, p01 = 0.05; (c) s = 0.2, p01OA = 0.03, p01OB = 0.06. The region A,O, B, AO, BO and AB correspond to theconnection probability p1, p0, p1, p01OA, p01OB and p11, respectively. The plots (a) and (b) show the symmetrical case of overlapping communities withp01OA = p01OB , and the last plot (c) is the asymmetrical case.

quite different from a general network, and can be applied to identify the overlapped group. Particularly, Li et al. analyticallyand numerically show that a complex network of phase oscillators may display interfaces between communities (clusters)of synchronized oscillations. The interfaces can be applied to identify the overlapped group of two communities [12];Almendral et al. provide a mechanism to extend the results of Ref. [12] from identifying the overlapped group of twocommunities to that of many communities [13].

In this paper, we further study the dynamics of networks with overlapping communities. For this purpose, a randomnetwork is generated first with a clear overlapping structure controlled by several parameters. Then, the synchronizationbehavior upon this network is investigated with various values of the controlling parameters. The purpose is to reveal whatare the most important factors (parameters) that affect the dynamical processes taking place upon the networks, and alsoshow how these factors affect the parallel synchronization dynamics in different communities and the overlapped group.In addition, the instantaneous frequencies of the communities and the overlapped group are also investigated in differenttime scales. The relationship between the dynamics (parallel synchronization behavior) and the overlapping structure canbe used to identify both the overlapping communities and the overlapped groups in networks.

2. Network model

Recently, some simple computer generated networks that have overlapping communities were introduced forcommunity detection [21]. Following these simple overlapping community networks, we introduce an extension of thosenetworks with overlapping communities to investigate its synchronization behavior.

For this purpose, we first construct an initial network G with size N by considering two large communities (A, B), andthe two large communities (A, B) are expected to be overlapped by a small group O with the overlapping size s (defined byEq. (1) in the following). Then connect every pair of nodes at random. The connection probability betweennodes belonging tothe larger community A/B is p1, particularly, p1A for A and p1B for B. The connection probability between nodes belonging tothe small overlapped group O is p0, which may be different from both p1A and p1B. The connection probability of two nodesbetween different communities (A and B) is p11, which is usuallymuch smaller than p1. The connection probability betweennodes in O and the rest nodes in A(B) is p01OA(p01OB). In real-word networks, p1A may be different from p1B; furthermore,the role of a node in O may be different from the rest nodes in A(B), it is not necessary that p01OA = p1A(p01OB = p1B),in these cases, p01OA and p01OB are different. It is worthwhile to point out that connection probability p01OA and p01OB areexpected to characterize the connectivity density of the overlapped groupO to the communities A and B, respectively. Hence,p01OA = p01OB and p01OA = p01OB correspond to the symmetrical and asymmetrical cases of overlapping, respectively. Inthe symmetrical case of overlapping, one can write p01OA = p01OB = p01 for convenience.

So far, a computer generated network is constructed with controllable overlapping, and the examples of overlappingcommunity networks (a), (b) and (c) can be seen in Fig. 1.

The parameter ‘overlapping size s’ is expected to describe the overlapping extent of communities, which is definedas

s =‖c1 ∩ c2‖‖c1 ∪ c2‖

, (1)

where, ‖c1 ∩ c2‖ denotes the number of nodes in both communities c1 and c2, and ‖c1 ∪ c2‖ denotes the combined sizeof c1 and c2. Apparently, the overlapping size of the proposed network is s = ‖O‖/N , where ‖O‖ is the number of nodes inO, and N is the size of the network.

For convenience, in the rest of this paper, community A(B) is used to denote the nodes in A(B) excluding those in O inthe case without confusion.

510 J. Wu et al. / Physica A 391 (2012) 508–514

a

b

Fig. 2. Order parameterM versus coupling strength c for different cases of parameter s and p01. All the data are obtained as the average over 50 realizations.

3. Phase synchronization on overlapping community networks

In this paper, the coupled phase oscillators, Kuramotomodel [26], is applied to analyze the dynamical behavior occurringon the proposed networks. The set of equations of motion governing the dynamics of the N oscillator system (i =

1, 2, . . . ,N) are written as:

dθidt

= ωi +cki

N−j=1

aij sin(θj − θi), (2)

where, θi and ωi are phase variables and intrinsic frequencies of node i, c is the coupling strength, and [aij] is the adjacencymatrix (aij = 1 if node i and j are connected; else aij = 0). In our simulation, the dynamical equations integrate using theRunge–Kutta method with step size 0.05. If there is no specific notification in the following, θi and ωi are randomly anduniformly distributed in the intervals [0, 2π) and [−0.5, 0.5] respectively.

To measure the synchronized states of the oscillators, an order parameterM is introduced:

M =

1NN−j=1

eiϕj

, (3)

where {·} denotes the time averaging. N is the number of nodes that are taken into account. Clearly, M is of order 1/√N if

the oscillators are completely uncoupled (c = 0), and will approach 1 if they are all in the same synchronized states.Here, in terms of the proposed network model, the networks with general parameters N = 100, p1 = 0.1, p11 = 0.005

are taken into account. Obviously, the size of both A and B is 45, and 10 for overlapped group O. Here, it is noticed that O isexcluded from A and B for clarity. Among these parameters, the relatively important parameters controlling the overlappingstructure are the overlapping size s, the connection probabilities p0, p01OA, and p01OB, which will be specially reported inthe following.

3.1. Synchronization with different overlapping parameters

How the synchronizability is affected by the overlapping size s is investigated first. Fig. 2(a) shows the simulation results.For all the different overlapping sizes s, the corresponding order parameters increase conformably with the increasing ofthe coupling strength c . In addition, for sufficiently large s (see the cases of s = 0.5 in Fig. 2(a)), the network can easily getsynchronized with a weak coupling strength; however, for sufficiently small s (see the cases of s = 0.005 in Fig. 2(a)), thenetwork is difficult to approach a completely synchronized state even with a very strong coupling strength. As a result, wecan come to a conclusion that the great overlapping size s will effectively enhance the synchronizability of the networks,indicating that, with greater overlapping size s (namely, more nodes of the overlapped group O), communities A and B canmore easily get harmonized with each other through O.

Second, how the synchronizability of the network is affected by the connection probabilities p0, p01OA, and p01OB isinvestigated. Here, without lack of generality, the symmetrical case of overlapping is taken into account, namely p01OA =

p01OB = p01. The purpose is to reveal what are the relatively important factors of overlapping structure that affect globaldynamical processes taking place upon the networks, and also show how these factors affect synchronization.

J. Wu et al. / Physica A 391 (2012) 508–514 511

Fig. 3. The correlation map of order parameter M in the two-dimensional parameter space (p0, p01) for the network. The value of M is represented bygray scale, and the coupling strength c = 5 is fixed.

Fig. 3 shows the correlation map of the order parameterM in the two-dimensional space (p0, p01) for the network withthe fixed coupling strength c = 5. It can be seen that the order parameters increase conformably only with the increasingof parameter p01 rather than p0, indicating the importance of the connection probability p01, namely the connectivitydensity of the overlapped group O to the communities A and B. Based on this observation, we further investigate how thesynchronizability of the networks is affected by the connection probability p01 with the coupling strength c , and Fig. 2(b)shows the simulation results. For all the different connection probability p01, the corresponding order parameters increasewith the increasing of the coupling strength c. In addition, for sufficiently large p01 (see the cases of p01 = 0.5 in Fig. 2(b)),the network can easily get synchronized with a weak coupling strength; however, for sufficiently small p01 (see the casesof p01 = 0.005 in Fig. 2(b)), the network is difficult to approach a completely synchronized state even with a very strongcoupling strength. And so, the great parameter p01will effectively enhance the synchronizability of the networks, indicatingthat, with greater connection probability p01, the communities A and B can more easily get harmonized with each otherthrough the community O with more links between them.

In addition, instead of observing the synchronizability of the network as a whole, we observe the parallel process(synchronization) occurring in A, B, and O, respectively. For the ith community (or group), similar to (3), the correspondingorder parameterMi is defined as

Mi =

1Nc

−j

eiϕj

, (4)

where i ∈ {A, B,O}, and the sum goes over all the nodes belonging to the ith community (or group).Based on (4), the temporal parallel behaviors of the order parameters in different topological scales, i.e. one community

(or group) and the whole network are investigated under different connection probabilities p01.Fig. 4 shows the simulation results, and the four plots correspond to the cases of p01 = 0.001, p01 = 0.01, p01 = 0.02,

and p01 = 0.1, respectively, corresponding to thicker and thicker connectivity density between O and the communitiesA and B. In the cases of p01 = 0.001, p01 = 0.01, and p01 = 0.02, the order parameters of the whole network and theoverlapped group O are much less than that of the communities A and B, and remarkably below 1 even in the long timelimit, indicating that the nodes in A and B are synchronized previous to that in O. This property can be used to detect boththe communities and the overlapped group. When p01 = 0.1, namely with sufficiently large p01, the links between O andA/B increase, then the overlapped group O itself not only can approach a nearly synchronized state, but also can harmonizewith other communities in some extent; therefore, the order parameter of A, B,O, and the whole network can approachthe nearly synchronized state almost as quickly as each other. This investigation reveals a remarkable relationship betweentopological scales and dynamic time scales: starting from random initial conditions, those highly interconnected nodes of thelarge communities A and Bwill synchronize first and then, in a sequential process, with the harmonization of the overlappedgroup O, the entire network will almost synchronized.

3.2. Instantaneous frequencies on synchronization

In this subsection, the instantaneous frequencies of A, B,O, and the whole network under different overlappingparameters are investigated.

512 J. Wu et al. / Physica A 391 (2012) 508–514

a b

c d

Fig. 4. The order parameter M versus time evolution with different parameters p01 for the whole network, communities A, B, and group O. The trianglecurve represents the order parameter of the whole network, and the square, circle, and diamond curves represent the order parameters of A, B and O,respectively. All the data are obtained as the average over 50 realizations, and the coupling strength c = 5 is fixed.

Set ωA = 0.7 ± 0.2, ωB = 0.3 ± 0.2, and ωO = 0.5 ± 0.5 (here, the initial intrinsic frequencies of communities A, B, andO are written as ωA, ωB, and ωO, respectively). Both the synchronization behaviors under symmetrical and asymmetricaloverlapping are addressed, respectively. The results are depicted in Fig. 5, where the instantaneous frequency of eachoscillator is a function of time. It is observed that the nodes in community A and B behave synchronously with a very smalloscillation around the mean of the original intrinsic frequency distribution, respectively. However, the nodes belonging tothe overlapped group O present a remarkable frequency with a very strong oscillation.

In the following, the relationship between the threemean frequencies of the communities (group) is analyzed. In the caseof plot (a), where the nodes of group O have the same probability (p01OA = p01OB = 0.01) to symmetrically connect to thenodes of the two large communities A and B, it is observed that the mean frequency of the nodes in group O is ωO = 0.4818,which is just approximately equal to ω = (ωA + ωB)/2 = 0.4910, where ωA = 0.6884 and ωB = 0.2935 are, respectively,the mean value of the frequencies of the two communities A and B. In the case of plot (b), where p01OA = 2p01OB = 0.02,indicating the overlapped group O has twofold links to community A. It is observed that the mean frequency of the nodesin O is ωO = 0.5616, which is approximately just a weighted mean frequency of the nodes in A and B, and shown asω = (2ωA + ωB)/3 = 0.5578, where ωA = 0.6756 and ωB = 0.3215. As for the case of plot (c), the overlapped groupO has twofold links to community B. It is observed that the mean frequency of the nodes in O is ωO = 0.4286, which is alsoa weighted mean frequency of ω = (ωA + 2ωB)/3 = 0.4438 approximately, where ωA = 0.7130 and ωB = 0.3092. Asa result, taking the different cases of plots (a), (b), and (c) into account, the mean frequency of the overlapped group O isrelevant to the connectivity density with communities A and B, and it is biased toward the community in which the nodeshavemore links. To be precise, it can be concluded that themean frequency of the overlapped group O tend to be aweightedmean frequency with respect to the communities A and B, and shown as:

ω =p01OAωA + p01OBωB

p01OA + p01OB(5)

where the connectivity probabilities p01OA and p01OB can be regarded as the weights with respect to the communities A andB. As a matter of fact, a similar result can also be found in Ref. [12], indicating the validity of our investigation.

In the case of plot (d), all the nodes of the network have the same initial distribution of frequencies in the range [0±0.5].Here, without lack of generality, the symmetrical case of overlapping is taken into account. It is observed that, starting fromthe oscillation in an early stage of time evolution, the mean frequency of the nodes in community A (or B) will approach toa nearly accordant state different from the mean frequency of B (or A), and that of group O possesses a mean value of thefrequencies of A and B, and then in a sequential process, the whole nodes in the network get into the same synchronizedstates with an almost constant frequency.

This investigation reveals a remarkable relationship of the frequencies of nodes in A, B,O, and the whole network in thesynchronization process. No matter the initial frequency distribution of the nodes in A, B, and O is the same or different, themean frequency of the nodes in A, B, and O behave differently in a given time scale. Especially in the case where all nodes of

J. Wu et al. / Physica A 391 (2012) 508–514 513

0 100 200 300

a

c

0 100 200 300

0 100 200 300 0 100 200 300

b

d

Fig. 5. Instantaneous frequencies θi(t) versus time evolution. The colors black, blue, red, and green refer respectively to the nodes belonging to the wholenetworks, A, B, and O, respectively. The general lines denote the instantaneous frequencies of the corresponding parts, and that the broken lines denote themean frequencies of the corresponding parts. In the plots (a), (b), and (c), the communities A, B, and group O have a different initial frequencies uniformlydistributed in the range [0.7 ± 0.2], [0.3 ± 0.2] and [0.5 ± 0.5], respectively. In the last plot (d), the initial frequencies of all nodes in the network areuniformly distributed in the range [0 ± 0.5]. The detailed parameters of the networks are: (a) p01OA = p01OB = 0.01, (b) p01OA = 2p01OB = 0.02, (c)2p01OA = p01OB = 0.02, (d) p01 = 0.01, and the other parameters are N = 100, s = 0.1, p1 = 0.1, p0 = 0.1, p11 = 0.001, and the coupling strengthc = 5 is fixed.

the network have the same initial distribution of frequencies, which has no need to detect the two communities previouslyand give them different distributions of initial frequencies, can be used to detect both the two communities A and B andtheir overlapped group O.

4. Conclusions and discussions

Based on the proposed network with overlapping community structure controlled by several parameters, a fundamentalrelationship between the overlapping community structure and the dynamical behavior occurring on the network has beeninvestigated. The most important factors (parameters) that affect the synchronization behavior are the overlapping size sand the connectivity density of the overlapped group O to the communities A and B. Increasing the overlapping size andconnectivity probability of O to A and B will effectively enhance the network synchronizability. In addition, it is found thatthe highly interconnected nodes in the communities A and B will synchronize first, and then in a sequential process, withthe harmonization of the overlapped groupO, the network evolve to the final state where all nodes are almost synchronized.

The finding in this paper can be potentially used to develop an algorithm to extract both the communities and theiroverlapped group in complex networks by dynamics toward synchronization, which requires further research.

In this paper, only the networks with two overlapping communities are investigated, but the method established can beextended to a real-word network with three (or more) pairwise overlapping communities.

514 J. Wu et al. / Physica A 391 (2012) 508–514

Acknowledgments

The authors thank the anonymous reviewers for their valuable comments and suggestions. This work was supported bytheNational Natural Science Foundation of China (Nos. 61072139, 61072106, and 61001202), and the Fundamental ResearchFunds for the Central Universities of China (Nos. Y10000902036, JY10000902039, and JY10000902001).

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