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A network evolution model based on community structure
Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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A network evolution model based on community structure Xue-Guang Wang a,b a b
Department of Computer Science, East China University of Political Science and Law, Shanghai 201620, China Department of Computer Science, University College London (UCL), London WC1E 6BT, UK
art ic l e i nf o
a b s t r a c t
Article history: Received 15 February 2015 Received in revised form 1 April 2015 Accepted 6 May 2015 Communicated by “Rongrong Ji”
Social networks are human-centered relationship communities. The research about social networks has an impact on the people’s daily life. In this paper, we observe the social networks based on the DBLP and Facebook datasets and confirm that the medium-scale community has a star-shaped structure and a core structure with high-connectivity and small diameter in the social networks. At the same time, we find that community merging depends largely on the clustering coefficient of the graph composed of nodes that connect two communities directly, and community splitting is largely decided by the clustering coefficient of this community. & 2015 Elsevier B.V. All rights reserved.
Keywords: Social networks Community evolution Network structure Network classification
1. Introduction A social network is a complicated structure composed of nodes and links. With the development of the Internet and the emergence of social networking sites, we can obtain the datasets of large-scale social networks from telecommunication operators or the Internet. According to the different functions, social networks can be simply classified into friendship networks (such as Facebook and MySpace), media-sharing networks (such as Youtube and Flickr), blogging networks (such as Livejournal and Twitter), IM networks (such as MSN and QQ), and BBS. All of these networks have tens of millions of registered users, which make them the perfect platform for our research on the structure and nature of social networks. During the 1960s, Stanley Milgram, a social psychologist at Harvard University, proposed the six degrees of separation theory. The idea behind this theory is that the average distance between any pair of people in the world is six degrees [1], which is considered as the theoretical basis of social networks. With the significant improvement of computing capacity, the development of the Internet and the interpenetration of different disciplines, we can manage and compare the data of large-scale social networks to determine their common structural features. Since Watts and Storagatz introduced the small-world network model in 1998 [2], and Barabási and Albert proposed the BA scale-free network model in 1999 [3], research on complex networks has attracted the attention of researchers from various fields such as mathematics, statistics, life sciences, information science, social sciences, and economics. Analyzing the social networks using the theory of complex networks focuses mainly on the correlations among the
nodes and their topological structures, which are also the foundation for determining the features and functions of social networks. Research on complex networks mainly considers its static and dynamic features. The former includes topological analysis [4,5], discovering key nodes [6–8], and so on. The latter mainly involves the formation and evolution of networks [9–12]. Considering community structure in a network, a number of researchers have proposed models for community generation, such as the copying model [13] and the JGN (Jin, Girvan, Newman) model [14]. Leskovec proposed the forest fire model to explain the structure of the community, diameter shrinking, and network densification [15]. By observing the time data of several social networks, a model was built according to the time sequence of the new nodes and edges joining the communities [10]. However, the evolution of the community structure is few researched. This paper study the evolution of social network based on complex network and community structure. To our knowledge, we have not found the same work. The contributions of our paper are as follows:
The feature and structure of social networks. We conducted
comprehensive research on the datasets and confirm that they have some properties, such as power-law distribution, densification law, and shrinking diameter. We found that the scale of the community in these networks obeys the power-law distribution and that the medium-scale communities have a starshaped structure. We also found a core structure with highconnectivity and small diameter in the social networks. The evolution of social networks. We conducted valuable research on the evolution of social networks based on event
http://dx.doi.org/10.1016/j.neucom.2015.05.021 0925-2312/& 2015 Elsevier B.V. All rights reserved.
Please cite this article as: X.-G. Wang, A network evolution model based on community structure, Neurocomputing (2015), http://dx. doi.org/10.1016/j.neucom.2015.05.021i
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framework. With the assistance of neural networks, we can identify the main structural factors that decide the emergence of events in the community. We found that community merging depends largely on the clustering coefficient of the graph composed of nodes that connect two communities directly, and community splitting is largely decided by the clustering coefficient of this community.
Fig. 1. An example of temporal and cumulative graphs.
The rest of this paper is organized as follows. Section 2 describes the dataset applied in this paper and the related major concepts. Section 3 confirms the features of social networks, and Section 4 presents their structural characteristics. Section 5 provides the structural conditions and factors of social network evolution. Our conclusions are provided in the last section.
concepts: average path length (APL), clustering coefficient, degree distribution [17], and assortativity coefficient [18]. In a network with N nodes, the distance dij between node i and node j is defined as the number of edges in the shortest path between them. The APL of the network is defined as
2. Datasets and related concepts
We assume that node i has ki neighbor nodes, that is, node i has degree ki . The ratio of the existing edge number ei among all these ki nodes to the theoretical maximum edge number ki ðki 1Þ=2 is
2.1. Datasets The datasets applied in this paper are from the DBLP computer science bibliography and the Facebook social networking site.
DBLP dataset
DBLP1, a tool used by researchers to search for computer science papers published in international journals and conferences, is highly reputable in academic circles. At present, over 1.4 million journals have been collected by DBLP and are available in XML format. We selected papers published in 80 important computer science journals from 1986 to 2006 as basic data, and treated the authors as nodes. An undirected edge was added between two nodes if two corresponding authors have written the same paper. In this manner, a DBLP co-authorship network that consists of 83,943 nodes and 190,342 edges is built. Facebook dataset Facebook is the most popular social networking site, which has over 500 million registered users. The data applied in this paper are from a previous report [16], which collected friendship linkages in New Orleans. We built a Facebook network with 60,567 nodes and 583,766 edges using data collected from 2007 to 2008.
2.2. Related concepts In general, for the social networks composed of nodes and edges, all of the corresponding data were obtained during the entire collection duration. Time information may be lost when the nodes interact with one another. Thus, we provide a dynamic model, which is a discrete time series of temporal graphs. The graph is formed by the nodes and edges obtained at specific time. The formal definition is as follows. Let Gt ¼ ðV t ; Et Þ denote the temporal graph at time t , V t represents a set of all the nodes at time t, and Et represents a set of all the edges appearing at t. The temporal graph sequence GR consists of G1 ; G2 ; …; GT ; if G0i ¼ ðV 0i ; E0i Þsuch that V 0i ¼ [ ik ¼ 1 V k andE0i ¼ [ ik ¼ 1 Ek , then G01 ; G02 ; …; G0T is called the cumulative graph sequenceGA . Let G ¼ G0T denote the final graph, an example of which is shown in Fig. 1. The other related papers provide numerous statistical properties of the network structure. Thus, we introduce only three basic 1
http://dblp.uni-trier.de/.
X 1 L¼1 d : i Z j ij 2NðN þ 1Þ
i defined as the clustering coefficient of node i, that is, C i ¼ ki ðk2e . i 1Þ P 1 Thus, the clustering coefficient of the network is C ¼ N i C i . The node degree distribution PðkÞ indicates the probability that the degree of a node chosen uniformly at random is k. The degree correlation denotes the average degree distributions of the neighborhood nodes for a specific node with a given degree, which can be characterized by the assortativity coefficient h i2 P P M 1 i ji ki M 1 i 12 ji þ ki r ¼ 1P 2 ; 1 rr r1: 2 1P 1 2 1 M i 2ðji þki Þ ½M i 2ðji þ ki Þ
where ji and ki are the degrees of the nodes at the ends of the ith edge with i ¼ 1; …; M, and M is the total number of all edges in the network. If r 4 0, the network is assortative; if r o0, the network is disassortative; and if r ¼ 0, no correlation exists among the node degrees.
3. The features of social networks In the theory of complex networks, one of the research objectives is to reveal the statistical features of network architecture and to determine how to measure them. Researchers found that networks have small-world and free-scale features, and they developed statistical methods such as degree distribution, APL, and clustering coefficient [19]. In this paper, we discuss the features of DBLP and Facebook from the standpoint of complex networks. Free-scale distribution widely exists in physics, geosciences, computer science, biology, demography, and economics α [20]. The formal definition is PðkÞ ¼ ck where k denotes a network feature variable, and c is a constant. For the degree distribution of nodes in the network, α generally lies in the range of 2 to 3 [21]. First, we discuss the degree distribution of the DBLP and Facebook datasets. These networks are undirected, and the influence of the “missing past” on network features is relatively minor [22]. Fig. 2 shows that the degree distributions of DBLP and Facebook are obviously scale-free, the power exponent of the former is 2.732, and the latter is 2.251. Their fitting precisions (R2) are 0.9967 and 0.9874, respectively. In addition, by using the CNM algorithm, we partitioned the communities of DBLP and Facebook, and calculated their respective sizes. We found that the community scale also obeys the power-law distribution, as shown in Fig. 3. Second, we discuss how the APL changes with time, and we characterize the changes based on the effective diameter [22]. If the distances of at least 90% of node pairs are smaller than d, the
Please cite this article as: X.-G. Wang, A network evolution model based on community structure, Neurocomputing (2015), http://dx. doi.org/10.1016/j.neucom.2015.05.021i
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minimum of d is defined as the effective diameter. As shown in Fig. 4, the effective diameter decreased over time. For DBLP, the journals were selected as data sources artificially. Numerous fields of computer science were at the early stage before 1990. Thus, the effective diameter changed significantly and declined gradually thereafter. For Facebook, the curve declined obviously at the start. The shrinking of the effective diameter is related to the density of the nodes and edges, as well as their increment rate. Thus, we discuss the changes of nodes and edges over time. Fig. 5 shows nonlinear growth from the edge number E to the node number N, which indicates the increase in average degree in the network.
Formally, it obeys scale-free distribution (E p N β ). In this manner, we confirm the densification laws and shrinking diameters of the DBLP and Facebook. For the former, β ¼ 1:285, R2 ¼ 0:9998, and for the latter, β ¼ 1:626, R2 ¼ 0:9987.
4. The structure of social networks In Section 3, we introduced the macro properties of social networks, and in this section, we discuss their micro structure. First, we explore how the community size of the network changes
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with time. Using the CNM algorithm, we partitioned the modules of the cumulative graphs of DBLP and Facebook and obtained the percentages of the communities with different sizes. The results are presented in Fig. 6. The bottom of the graph represents a small-scale community (owns a few nodes), the middle part represents a medium-scale community (owns tens to hundreds of nodes), and the top represents a large-scale community (owns thousands of nodes). During the formation of the networks, the percentage of the medium-scale community gradually decreases over time, whereas the large-scale community increases and the small-scale community basically remain unchanged. Moreover, the assortativity coefficient of the medium-scale community is almost disassortative, which indicates that nodes with larger degree are inclined to connect with those with a smaller degree. Therefore,
most of medium-scale communities in these two networks have an obvious star-shaped structure. Second, we discuss whether cores with small diameter and high connectivity are observed in social networks. We removed the nodes from those of smaller degree to larger degree gradually and recorded the changes in effective diameter and the ratio of edge to node. The results are shown in Fig. 7. Fig. 7(a) indicates that the effective diameter of DBLP decreased by 28.94% and that of Facebook by 21.85%. On the contrary, their edge-to-node ratio increased to 3.28 times and 1.88 times, respectively, as reported in Fig. 7(b). Based on these results, we can conclude that core structures exist in both networks. However, these findings do not imply that the conductance of the core structure is the best. For a node set, the ratio of the
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number of edges with one end inside the set and the other end outside the set to the total number of the edges in the set is defined as its conductance. However, obtaining the subset with k nodes and the weakest conductance presents a difficult problem. Using the network community profile (NCP) method [4], we considered the conductance of the communities with different k in DBLP and Facebook. As shown in Fig. 8, the node number of a community with good conductance ranges between 10 and 100. The conductance becomes weak with the increase in community size. By seeking out the communities with weak conductance, we found that most of them were on the boundary of the network, with few links to other communities.
5. The evolution of social networks Asur et al. presented an event-based framework to describe the relationship between communities, but they concentrated mainly on the stability, influence, and sociability of individuals as well as the popularity of the community [23]. Based on this framework, our research focused on the interaction among communities. In the following paragraphs, we provide the details of the framework. 5.1. Event-based framework An event is defined as the impact of the interaction among communities in graph Gt and graph Gt þ 1 , which includes merging, splitting, continuation, formation, and dissolution. We determined whether the events occur based on changes in the number of nodes in the same community, and the changes in connectivity among these nodes are not considered if the node number remains unchanged. We assume that graph Gt ¼ ðV t ; Et Þ is partitioned into m comi i i munities fC 1t ; C 2t ; …; C m t g where the ith community C t ¼ ðV t ; E t Þ and i i V t D V t , Et D Et . The symbol j j denotes the cardinal of a set and φ ¼ 0:5. If no community C it is found in Gt such that j V it \ V ktþ 1 j 41, a community is formed in Gt þ 1 . If no community C ktþ 1 is found in Gt þ 1 such that k j V t þ 1 \ V it j 41, a community is dissolved in Gt þ 1 . If communities C it are found in Gt and C ktþ 1 in Gt þ 1 such that V it ¼ V ktþ 1 , a community is continued in Gt þ 1 . If communities C it and C jt in are found in Gt , and C ktþ 1 are found in Gt þ 1 such that j ðV it [ V jt Þ \ V ktþ 1 j Maxðj V it [ V jt j ; j V ktþ 1 j Þ
4 φ;
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j Ci j
j V it \ V ktþ 1 j 4 2t and j V jt \ V ktþ 1 j 4 2t , merging occurs between C it and C jt in Gt þ 1 . If communities C it are found in Gt , and C jt þ 1 and C ktþ 1 are found in Gt þ 1 such that j ðV jt þ 1 [ V ktþ 1 Þ \ V it j Maxðj V jt þ 1 [ V ktþ 1 j ; j V it j Þ j Cj
j
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j V jt þ 1 \ V it j 4 t2þ 1 and j V ktþ 1 \ V it j 4 t2þ 1 , a split of C it occurs in Gt þ 1 . The structural features of community evolution Community dynamics is one of the key research subjects in social network analysis. We are interested in identifying the type of structures that affect the evolution of a community. We take the following factors into account. Feature 1: Average distance between two communities, which is defined as the average of the shortest path lengths of all node pairs between these two communities. Feature 2: The number of nodes that directly connect two communities and the clustering coefficient of the graph formed by these nodes. Feature 3: The number of nodes in one community that directly connects to the other community, and the clustering coefficient of the graph formed by these nodes. Feature 4: The number of nodes in one community and the clustering coefficient of this community.
To determine which factor has a main function in community evolution, these data were inputted into the BP neural networks. The events were partitioned into training samples and test samples during the evolution. By controlling different factors as input, several neural networks were built. The significance of these factors was estimated according to the prediction accuracy of the corresponding neural network. All the neural networks have the same three-tier architecture and use the LM learning algorithm. The input tier has nine neurons that correspond to the nine different structure characteristics. The number of neurons in the hidden tier is 25, which is obtained by a large number of trials, and the transfer function used is sigmoid. The output tier has one neuron, which has a linear function. To assess the prediction accuracy of these neural networks, we used the receiver operation characteristic (ROC) curves, which consider both sensitivity and 1-specificity. The area under the ROC curve (AUC) facilitated the comparison among the performance of different classifiers. We focused on the feature factors of community merging and splitting. For DBLP and Facebook, we obtained
Please cite this article as: X.-G. Wang, A network evolution model based on community structure, Neurocomputing (2015), http://dx. doi.org/10.1016/j.neucom.2015.05.021i
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consistent conclusions. Thus, we provide only the experimental results of DBLP in this paper.
Table 1 AUC with different features. AUC
Community merging Our findings indicate that from 1996 to 2006, the number of communities in DBLP such that the constraints j V it \ j Cj j j Ci j V ktþ 1 j 4 2t and j V jt \ V ktþ 1 j 4 2t is 5722 pairs, and 768 of them merged. Based on these data, we used 4000 pairs (including the 550 merged pairs) as the training samples and the others as test samples to build four neural networks for prediction. The prediction accuracy is reported in Fig. 9. Fig. 9 shows that the different features have different influences on community merging. The AURs of Features 4, 3, and 2 are 0.5563, 0.6922, and 0.7150, whereas the AUR is 0.8349 when all of the features were used. A more convex ROC curve indicates a larger AUR, which results in a more accurate prediction. As shown in Fig. 11, we believe that Feature 2 has a more significant function in community evolution. To validate the conclusion further, we determined the 3751 pairs with direct linkages from 5722 pairs of communities (including 409 pairs of merged communities). We calculated the clustering coefficients of the graphs formed by nodes and edges directly connecting two communities. Then, the numbers of directly connected community pairs and merged community pairs with clustering coefficients that are respectively located within the 10 equidistant intervals between 0 and 1 were counted. The ratios of these values were taken as the approximate probabilities of community merging. Fig. 10 shows that when the
Feature 2 Feature 3 Feature 4 All features
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clustering coefficient between two communities increases, the probability of community merging becomes larger. Community splitting From 1996 to 2006, the number of communities in DBLP such j C jt þ 1 j 2
and j V ktþ 1 \ V it j
j Ck j 4 t2þ 1
is 3811 pairs, in which 475 pairs are generated by community splitting. Using neural networks for prediction, the training sample includes 2500 pairs (300 of which resulted from community splitting) and the others are test samples. The results are presented in Table 1.
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As the table indicates, Feature 4 is more accurate for prediction. We also studied the relationship between clustering coefficient and community splitting. The results in Fig. 11 indicate that a larger clustering coefficient indicates greater difficulty in splitting the community.
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6. Conclusions and future works In this paper we introduce our research on the characteristics of social networks and the structure features of the community evolution. We found that the medium-scale community has a starshaped structure and a core structure with high-connectivity and small diameter in the social networks. We conducted valuable research on the evolution of social networks based on event framework. We found that community merging depends largely on the clustering coefficient of the graph composed of nodes that connect two communities directly, and community splitting is largely decided by the clustering coefficient of this community. The social networks are human-centered relationship networks and the research in this field will impact on people’s daily life. The development of the internet and the emergence of the social network sites provide a perfect platform for our research. How are the natures of social networks formed? Is there a theory model which can be able to explain the natures appearing during the interaction of the nodes? What influence exists between different
Please cite this article as: X.-G. Wang, A network evolution model based on community structure, Neurocomputing (2015), http://dx. doi.org/10.1016/j.neucom.2015.05.021i
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network topologies and individual behavior? How to characterize and control the information spreading on social networks? These are the problems for us to further research and resolve. Acknowledgements This work is supported by National Social Science Foundation of China (no. 11BFX125), PuJiang Talent Project and Public Security Discipline Construction Foundation.
References [1] S. Milgram, The small world problem,, Psychol. Today 2 (1) (1967) 60–67. [2] D. Watts, S. Strogatz, Collective dynamics of small world networks, Nature 393 (6684) (1998) 440–442. [3] A. Barabási, R. Albert, Emergence of scaling in random networks, Science 286 (5439) (1999) 509. [4] J. Leskovec, K. Lang, A. Dasgupta, M. Mahoney, Statistical properties of community structure in large social and information networks, in: Proceeding of the 17th International Conference on World Wide Web, ACM. , 2008, pp. 695–704. [5] A. Mislove, M. Marcon, K. Gummadi, P. Druschel, B. Bhattacharjee, Measurement and analysis of online social networks, in: Proceedings of the Seventh ACM SIGCOMM Conference on Internet Measurement. , 2007, pp. 29–42. [6] T. Guan, Y. He, J. Gao, J. Yang, J. Yu, On-device mobile visual location recognition by integrating vision and inertial sensors, IEEE Trans. Multimedia 15 (7) (2013) 1688–1699. [7] S. White, P. Smyth, Algorithms for estimating relative importance in networks, in: Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. , 2003, pp. 266–275. [8] Y. Li, C. Lai, C. Chen, Identifying bloggers with marketing influence in the blogosphere, in: Proceedings of the 11th International Conference on Electronic Commerce, ACM. , 2009, pp. 335–340. [9] T. Guan, Y.F. He, L.Y. Duan, J.Q. Yu, Efficient BOF generation and compression for on-device mobile visual location recognition, IEEE Multimedia 21 (2) (2014) 32–41. [10] J. Leskovec, L. Backstrom, R. Kumar, A. Tomkins, Microscopic evolution of social networks, in: Proceeding of the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. , 2008, 462–470. [11] Y. Lin, Y. Chi, S. Zhu, H. Sundaram, B. Tseng, Analyzing communities and their evolutions in dynamic social networks, ACM Trans. Knowledge Discov. Data (TKDD) 3 (2) (2009) 1–31. [12] Benchang Wei, T.a.o. Guan, Junqing Yu, Projected residual vector quantization for ANN search, IEEE Multimedia 21 (3) (2014) 41–51.
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[13] R. Kumar, P. Raghavan, S. Rajagopalan, D. Sivakumar, A. Tomkins, E. Upfal, Stochastic models for the web graph, in: . Foundations of Computer Science, 2000. IEEE Proceedings. 41st Annual Symposium on. ; 2002, pp. 57–65. [14] E. Jin, M. Girvan, M. Newman, Structure of growing social networks, Phys. Rev. E 64 (4) (2001) 46132. [15] X. Li, G. Chen, A local-world evolving network model, Physica A: Stat. Mech. Appl. 328 (1–2) (2003) 274–286. [16] B. Viswanath, A. Mislove, M. Cha, K. Gummadi, On the evolution of user interaction in facebook, in: Proceedings of the Second ACM Workshop on Online Social Networks, ACM. ; 2009, pp. 37–42. [17] X.F. Wang, X. Li, G.R. Chen, Theory and Application of Complex Networks, Tsinghua University Press, Beijing, 2006. [18] M. Newman, Assortative mixing in networks, Phys. Rev. Lett. 89 (20) (2002) 208701. [19] F. Rodrigues, G. Travieso, V. Boas, Characterization of complex networks: a survey of measurements, Adv. Phys. 56 (1) (2007) 167–242. [20] H.B. Hu, L. Wang, A brief history of power law distributions, Physics 34 (012) (2005) 889–896. [21] R. Albert, A. Barabási, Statistical mechanics of complex networks,, Rev. Mod. Phys. 74 (no. 1) (2002) 47–97. [22] J. Leskovec, J. Kleinberg, C. Faloutsos, Graphs over time: densification laws, shrinking diameters and possible explanations, in: Proceedings of the 11th ACM SIGKDD International Conference on Knowledge Discovery in Data Mining. , 2005, pp. 177–187. [23] S. Asur, S. Parthasarathy, D. Ucar, An event-based framework for characterizing the evolutionary behavior of interaction graphs, ACM Trans. Knowledge Discov. Data (TKDD) 3 (4) (2009) 1–36.
Xueguang Wang is professor of East China University of Political Science and Law. Visiting Academic at University College London (UCL, UK). He received the PhD degree in Computer Networks at East China University of Science and Technology. His main research interests are: computer networks, information security and electronic evidence. He has published research articles in scientific world journal and Applied Mathematics & Information Sciences.
Please cite this article as: X.-G. Wang, A network evolution model based on community structure, Neurocomputing (2015), http://dx. doi.org/10.1016/j.neucom.2015.05.021i
Contents lists available at ScienceDirect
Neurocomputing journal homepage: www.elsevier.com/locate/neucom
A network evolution model based on community structure Xue-Guang Wang a,b a b
Department of Computer Science, East China University of Political Science and Law, Shanghai 201620, China Department of Computer Science, University College London (UCL), London WC1E 6BT, UK
art ic l e i nf o
a b s t r a c t
Article history: Received 15 February 2015 Received in revised form 1 April 2015 Accepted 6 May 2015 Communicated by “Rongrong Ji”
Social networks are human-centered relationship communities. The research about social networks has an impact on the people’s daily life. In this paper, we observe the social networks based on the DBLP and Facebook datasets and confirm that the medium-scale community has a star-shaped structure and a core structure with high-connectivity and small diameter in the social networks. At the same time, we find that community merging depends largely on the clustering coefficient of the graph composed of nodes that connect two communities directly, and community splitting is largely decided by the clustering coefficient of this community. & 2015 Elsevier B.V. All rights reserved.
Keywords: Social networks Community evolution Network structure Network classification
1. Introduction A social network is a complicated structure composed of nodes and links. With the development of the Internet and the emergence of social networking sites, we can obtain the datasets of large-scale social networks from telecommunication operators or the Internet. According to the different functions, social networks can be simply classified into friendship networks (such as Facebook and MySpace), media-sharing networks (such as Youtube and Flickr), blogging networks (such as Livejournal and Twitter), IM networks (such as MSN and QQ), and BBS. All of these networks have tens of millions of registered users, which make them the perfect platform for our research on the structure and nature of social networks. During the 1960s, Stanley Milgram, a social psychologist at Harvard University, proposed the six degrees of separation theory. The idea behind this theory is that the average distance between any pair of people in the world is six degrees [1], which is considered as the theoretical basis of social networks. With the significant improvement of computing capacity, the development of the Internet and the interpenetration of different disciplines, we can manage and compare the data of large-scale social networks to determine their common structural features. Since Watts and Storagatz introduced the small-world network model in 1998 [2], and Barabási and Albert proposed the BA scale-free network model in 1999 [3], research on complex networks has attracted the attention of researchers from various fields such as mathematics, statistics, life sciences, information science, social sciences, and economics. Analyzing the social networks using the theory of complex networks focuses mainly on the correlations among the
nodes and their topological structures, which are also the foundation for determining the features and functions of social networks. Research on complex networks mainly considers its static and dynamic features. The former includes topological analysis [4,5], discovering key nodes [6–8], and so on. The latter mainly involves the formation and evolution of networks [9–12]. Considering community structure in a network, a number of researchers have proposed models for community generation, such as the copying model [13] and the JGN (Jin, Girvan, Newman) model [14]. Leskovec proposed the forest fire model to explain the structure of the community, diameter shrinking, and network densification [15]. By observing the time data of several social networks, a model was built according to the time sequence of the new nodes and edges joining the communities [10]. However, the evolution of the community structure is few researched. This paper study the evolution of social network based on complex network and community structure. To our knowledge, we have not found the same work. The contributions of our paper are as follows:
The feature and structure of social networks. We conducted
comprehensive research on the datasets and confirm that they have some properties, such as power-law distribution, densification law, and shrinking diameter. We found that the scale of the community in these networks obeys the power-law distribution and that the medium-scale communities have a starshaped structure. We also found a core structure with highconnectivity and small diameter in the social networks. The evolution of social networks. We conducted valuable research on the evolution of social networks based on event
http://dx.doi.org/10.1016/j.neucom.2015.05.021 0925-2312/& 2015 Elsevier B.V. All rights reserved.
Please cite this article as: X.-G. Wang, A network evolution model based on community structure, Neurocomputing (2015), http://dx. doi.org/10.1016/j.neucom.2015.05.021i
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framework. With the assistance of neural networks, we can identify the main structural factors that decide the emergence of events in the community. We found that community merging depends largely on the clustering coefficient of the graph composed of nodes that connect two communities directly, and community splitting is largely decided by the clustering coefficient of this community.
Fig. 1. An example of temporal and cumulative graphs.
The rest of this paper is organized as follows. Section 2 describes the dataset applied in this paper and the related major concepts. Section 3 confirms the features of social networks, and Section 4 presents their structural characteristics. Section 5 provides the structural conditions and factors of social network evolution. Our conclusions are provided in the last section.
concepts: average path length (APL), clustering coefficient, degree distribution [17], and assortativity coefficient [18]. In a network with N nodes, the distance dij between node i and node j is defined as the number of edges in the shortest path between them. The APL of the network is defined as
2. Datasets and related concepts
We assume that node i has ki neighbor nodes, that is, node i has degree ki . The ratio of the existing edge number ei among all these ki nodes to the theoretical maximum edge number ki ðki 1Þ=2 is
2.1. Datasets The datasets applied in this paper are from the DBLP computer science bibliography and the Facebook social networking site.
DBLP dataset
DBLP1, a tool used by researchers to search for computer science papers published in international journals and conferences, is highly reputable in academic circles. At present, over 1.4 million journals have been collected by DBLP and are available in XML format. We selected papers published in 80 important computer science journals from 1986 to 2006 as basic data, and treated the authors as nodes. An undirected edge was added between two nodes if two corresponding authors have written the same paper. In this manner, a DBLP co-authorship network that consists of 83,943 nodes and 190,342 edges is built. Facebook dataset Facebook is the most popular social networking site, which has over 500 million registered users. The data applied in this paper are from a previous report [16], which collected friendship linkages in New Orleans. We built a Facebook network with 60,567 nodes and 583,766 edges using data collected from 2007 to 2008.
2.2. Related concepts In general, for the social networks composed of nodes and edges, all of the corresponding data were obtained during the entire collection duration. Time information may be lost when the nodes interact with one another. Thus, we provide a dynamic model, which is a discrete time series of temporal graphs. The graph is formed by the nodes and edges obtained at specific time. The formal definition is as follows. Let Gt ¼ ðV t ; Et Þ denote the temporal graph at time t , V t represents a set of all the nodes at time t, and Et represents a set of all the edges appearing at t. The temporal graph sequence GR consists of G1 ; G2 ; …; GT ; if G0i ¼ ðV 0i ; E0i Þsuch that V 0i ¼ [ ik ¼ 1 V k andE0i ¼ [ ik ¼ 1 Ek , then G01 ; G02 ; …; G0T is called the cumulative graph sequenceGA . Let G ¼ G0T denote the final graph, an example of which is shown in Fig. 1. The other related papers provide numerous statistical properties of the network structure. Thus, we introduce only three basic 1
http://dblp.uni-trier.de/.
X 1 L¼1 d : i Z j ij 2NðN þ 1Þ
i defined as the clustering coefficient of node i, that is, C i ¼ ki ðk2e . i 1Þ P 1 Thus, the clustering coefficient of the network is C ¼ N i C i . The node degree distribution PðkÞ indicates the probability that the degree of a node chosen uniformly at random is k. The degree correlation denotes the average degree distributions of the neighborhood nodes for a specific node with a given degree, which can be characterized by the assortativity coefficient h i2 P P M 1 i ji ki M 1 i 12 ji þ ki r ¼ 1P 2 ; 1 rr r1: 2 1P 1 2 1 M i 2ðji þki Þ ½M i 2ðji þ ki Þ
where ji and ki are the degrees of the nodes at the ends of the ith edge with i ¼ 1; …; M, and M is the total number of all edges in the network. If r 4 0, the network is assortative; if r o0, the network is disassortative; and if r ¼ 0, no correlation exists among the node degrees.
3. The features of social networks In the theory of complex networks, one of the research objectives is to reveal the statistical features of network architecture and to determine how to measure them. Researchers found that networks have small-world and free-scale features, and they developed statistical methods such as degree distribution, APL, and clustering coefficient [19]. In this paper, we discuss the features of DBLP and Facebook from the standpoint of complex networks. Free-scale distribution widely exists in physics, geosciences, computer science, biology, demography, and economics α [20]. The formal definition is PðkÞ ¼ ck where k denotes a network feature variable, and c is a constant. For the degree distribution of nodes in the network, α generally lies in the range of 2 to 3 [21]. First, we discuss the degree distribution of the DBLP and Facebook datasets. These networks are undirected, and the influence of the “missing past” on network features is relatively minor [22]. Fig. 2 shows that the degree distributions of DBLP and Facebook are obviously scale-free, the power exponent of the former is 2.732, and the latter is 2.251. Their fitting precisions (R2) are 0.9967 and 0.9874, respectively. In addition, by using the CNM algorithm, we partitioned the communities of DBLP and Facebook, and calculated their respective sizes. We found that the community scale also obeys the power-law distribution, as shown in Fig. 3. Second, we discuss how the APL changes with time, and we characterize the changes based on the effective diameter [22]. If the distances of at least 90% of node pairs are smaller than d, the
Please cite this article as: X.-G. Wang, A network evolution model based on community structure, Neurocomputing (2015), http://dx. doi.org/10.1016/j.neucom.2015.05.021i
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minimum of d is defined as the effective diameter. As shown in Fig. 4, the effective diameter decreased over time. For DBLP, the journals were selected as data sources artificially. Numerous fields of computer science were at the early stage before 1990. Thus, the effective diameter changed significantly and declined gradually thereafter. For Facebook, the curve declined obviously at the start. The shrinking of the effective diameter is related to the density of the nodes and edges, as well as their increment rate. Thus, we discuss the changes of nodes and edges over time. Fig. 5 shows nonlinear growth from the edge number E to the node number N, which indicates the increase in average degree in the network.
Formally, it obeys scale-free distribution (E p N β ). In this manner, we confirm the densification laws and shrinking diameters of the DBLP and Facebook. For the former, β ¼ 1:285, R2 ¼ 0:9998, and for the latter, β ¼ 1:626, R2 ¼ 0:9987.
4. The structure of social networks In Section 3, we introduced the macro properties of social networks, and in this section, we discuss their micro structure. First, we explore how the community size of the network changes
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with time. Using the CNM algorithm, we partitioned the modules of the cumulative graphs of DBLP and Facebook and obtained the percentages of the communities with different sizes. The results are presented in Fig. 6. The bottom of the graph represents a small-scale community (owns a few nodes), the middle part represents a medium-scale community (owns tens to hundreds of nodes), and the top represents a large-scale community (owns thousands of nodes). During the formation of the networks, the percentage of the medium-scale community gradually decreases over time, whereas the large-scale community increases and the small-scale community basically remain unchanged. Moreover, the assortativity coefficient of the medium-scale community is almost disassortative, which indicates that nodes with larger degree are inclined to connect with those with a smaller degree. Therefore,
most of medium-scale communities in these two networks have an obvious star-shaped structure. Second, we discuss whether cores with small diameter and high connectivity are observed in social networks. We removed the nodes from those of smaller degree to larger degree gradually and recorded the changes in effective diameter and the ratio of edge to node. The results are shown in Fig. 7. Fig. 7(a) indicates that the effective diameter of DBLP decreased by 28.94% and that of Facebook by 21.85%. On the contrary, their edge-to-node ratio increased to 3.28 times and 1.88 times, respectively, as reported in Fig. 7(b). Based on these results, we can conclude that core structures exist in both networks. However, these findings do not imply that the conductance of the core structure is the best. For a node set, the ratio of the
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number of edges with one end inside the set and the other end outside the set to the total number of the edges in the set is defined as its conductance. However, obtaining the subset with k nodes and the weakest conductance presents a difficult problem. Using the network community profile (NCP) method [4], we considered the conductance of the communities with different k in DBLP and Facebook. As shown in Fig. 8, the node number of a community with good conductance ranges between 10 and 100. The conductance becomes weak with the increase in community size. By seeking out the communities with weak conductance, we found that most of them were on the boundary of the network, with few links to other communities.
5. The evolution of social networks Asur et al. presented an event-based framework to describe the relationship between communities, but they concentrated mainly on the stability, influence, and sociability of individuals as well as the popularity of the community [23]. Based on this framework, our research focused on the interaction among communities. In the following paragraphs, we provide the details of the framework. 5.1. Event-based framework An event is defined as the impact of the interaction among communities in graph Gt and graph Gt þ 1 , which includes merging, splitting, continuation, formation, and dissolution. We determined whether the events occur based on changes in the number of nodes in the same community, and the changes in connectivity among these nodes are not considered if the node number remains unchanged. We assume that graph Gt ¼ ðV t ; Et Þ is partitioned into m comi i i munities fC 1t ; C 2t ; …; C m t g where the ith community C t ¼ ðV t ; E t Þ and i i V t D V t , Et D Et . The symbol j j denotes the cardinal of a set and φ ¼ 0:5. If no community C it is found in Gt such that j V it \ V ktþ 1 j 41, a community is formed in Gt þ 1 . If no community C ktþ 1 is found in Gt þ 1 such that k j V t þ 1 \ V it j 41, a community is dissolved in Gt þ 1 . If communities C it are found in Gt and C ktþ 1 in Gt þ 1 such that V it ¼ V ktþ 1 , a community is continued in Gt þ 1 . If communities C it and C jt in are found in Gt , and C ktþ 1 are found in Gt þ 1 such that j ðV it [ V jt Þ \ V ktþ 1 j Maxðj V it [ V jt j ; j V ktþ 1 j Þ
4 φ;
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j V it \ V ktþ 1 j 4 2t and j V jt \ V ktþ 1 j 4 2t , merging occurs between C it and C jt in Gt þ 1 . If communities C it are found in Gt , and C jt þ 1 and C ktþ 1 are found in Gt þ 1 such that j ðV jt þ 1 [ V ktþ 1 Þ \ V it j Maxðj V jt þ 1 [ V ktþ 1 j ; j V it j Þ j Cj
j
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j V jt þ 1 \ V it j 4 t2þ 1 and j V ktþ 1 \ V it j 4 t2þ 1 , a split of C it occurs in Gt þ 1 . The structural features of community evolution Community dynamics is one of the key research subjects in social network analysis. We are interested in identifying the type of structures that affect the evolution of a community. We take the following factors into account. Feature 1: Average distance between two communities, which is defined as the average of the shortest path lengths of all node pairs between these two communities. Feature 2: The number of nodes that directly connect two communities and the clustering coefficient of the graph formed by these nodes. Feature 3: The number of nodes in one community that directly connects to the other community, and the clustering coefficient of the graph formed by these nodes. Feature 4: The number of nodes in one community and the clustering coefficient of this community.
To determine which factor has a main function in community evolution, these data were inputted into the BP neural networks. The events were partitioned into training samples and test samples during the evolution. By controlling different factors as input, several neural networks were built. The significance of these factors was estimated according to the prediction accuracy of the corresponding neural network. All the neural networks have the same three-tier architecture and use the LM learning algorithm. The input tier has nine neurons that correspond to the nine different structure characteristics. The number of neurons in the hidden tier is 25, which is obtained by a large number of trials, and the transfer function used is sigmoid. The output tier has one neuron, which has a linear function. To assess the prediction accuracy of these neural networks, we used the receiver operation characteristic (ROC) curves, which consider both sensitivity and 1-specificity. The area under the ROC curve (AUC) facilitated the comparison among the performance of different classifiers. We focused on the feature factors of community merging and splitting. For DBLP and Facebook, we obtained
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consistent conclusions. Thus, we provide only the experimental results of DBLP in this paper.
Table 1 AUC with different features. AUC
Community merging Our findings indicate that from 1996 to 2006, the number of communities in DBLP such that the constraints j V it \ j Cj j j Ci j V ktþ 1 j 4 2t and j V jt \ V ktþ 1 j 4 2t is 5722 pairs, and 768 of them merged. Based on these data, we used 4000 pairs (including the 550 merged pairs) as the training samples and the others as test samples to build four neural networks for prediction. The prediction accuracy is reported in Fig. 9. Fig. 9 shows that the different features have different influences on community merging. The AURs of Features 4, 3, and 2 are 0.5563, 0.6922, and 0.7150, whereas the AUR is 0.8349 when all of the features were used. A more convex ROC curve indicates a larger AUR, which results in a more accurate prediction. As shown in Fig. 11, we believe that Feature 2 has a more significant function in community evolution. To validate the conclusion further, we determined the 3751 pairs with direct linkages from 5722 pairs of communities (including 409 pairs of merged communities). We calculated the clustering coefficients of the graphs formed by nodes and edges directly connecting two communities. Then, the numbers of directly connected community pairs and merged community pairs with clustering coefficients that are respectively located within the 10 equidistant intervals between 0 and 1 were counted. The ratios of these values were taken as the approximate probabilities of community merging. Fig. 10 shows that when the
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clustering coefficient between two communities increases, the probability of community merging becomes larger. Community splitting From 1996 to 2006, the number of communities in DBLP such j C jt þ 1 j 2
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Clustering coefficient Fig. 10. Influence of clustering coefficient on community merging.
6. Conclusions and future works In this paper we introduce our research on the characteristics of social networks and the structure features of the community evolution. We found that the medium-scale community has a starshaped structure and a core structure with high-connectivity and small diameter in the social networks. We conducted valuable research on the evolution of social networks based on event framework. We found that community merging depends largely on the clustering coefficient of the graph composed of nodes that connect two communities directly, and community splitting is largely decided by the clustering coefficient of this community. The social networks are human-centered relationship networks and the research in this field will impact on people’s daily life. The development of the internet and the emergence of the social network sites provide a perfect platform for our research. How are the natures of social networks formed? Is there a theory model which can be able to explain the natures appearing during the interaction of the nodes? What influence exists between different
Please cite this article as: X.-G. Wang, A network evolution model based on community structure, Neurocomputing (2015), http://dx. doi.org/10.1016/j.neucom.2015.05.021i
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network topologies and individual behavior? How to characterize and control the information spreading on social networks? These are the problems for us to further research and resolve. Acknowledgements This work is supported by National Social Science Foundation of China (no. 11BFX125), PuJiang Talent Project and Public Security Discipline Construction Foundation.
References [1] S. Milgram, The small world problem,, Psychol. Today 2 (1) (1967) 60–67. [2] D. Watts, S. Strogatz, Collective dynamics of small world networks, Nature 393 (6684) (1998) 440–442. [3] A. Barabási, R. Albert, Emergence of scaling in random networks, Science 286 (5439) (1999) 509. [4] J. Leskovec, K. Lang, A. Dasgupta, M. Mahoney, Statistical properties of community structure in large social and information networks, in: Proceeding of the 17th International Conference on World Wide Web, ACM. , 2008, pp. 695–704. [5] A. Mislove, M. Marcon, K. Gummadi, P. Druschel, B. Bhattacharjee, Measurement and analysis of online social networks, in: Proceedings of the Seventh ACM SIGCOMM Conference on Internet Measurement. , 2007, pp. 29–42. [6] T. Guan, Y. He, J. Gao, J. Yang, J. Yu, On-device mobile visual location recognition by integrating vision and inertial sensors, IEEE Trans. Multimedia 15 (7) (2013) 1688–1699. [7] S. White, P. Smyth, Algorithms for estimating relative importance in networks, in: Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. , 2003, pp. 266–275. [8] Y. Li, C. Lai, C. Chen, Identifying bloggers with marketing influence in the blogosphere, in: Proceedings of the 11th International Conference on Electronic Commerce, ACM. , 2009, pp. 335–340. [9] T. Guan, Y.F. He, L.Y. Duan, J.Q. Yu, Efficient BOF generation and compression for on-device mobile visual location recognition, IEEE Multimedia 21 (2) (2014) 32–41. [10] J. Leskovec, L. Backstrom, R. Kumar, A. Tomkins, Microscopic evolution of social networks, in: Proceeding of the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. , 2008, 462–470. [11] Y. Lin, Y. Chi, S. Zhu, H. Sundaram, B. Tseng, Analyzing communities and their evolutions in dynamic social networks, ACM Trans. Knowledge Discov. Data (TKDD) 3 (2) (2009) 1–31. [12] Benchang Wei, T.a.o. Guan, Junqing Yu, Projected residual vector quantization for ANN search, IEEE Multimedia 21 (3) (2014) 41–51.
7
[13] R. Kumar, P. Raghavan, S. Rajagopalan, D. Sivakumar, A. Tomkins, E. Upfal, Stochastic models for the web graph, in: . Foundations of Computer Science, 2000. IEEE Proceedings. 41st Annual Symposium on. ; 2002, pp. 57–65. [14] E. Jin, M. Girvan, M. Newman, Structure of growing social networks, Phys. Rev. E 64 (4) (2001) 46132. [15] X. Li, G. Chen, A local-world evolving network model, Physica A: Stat. Mech. Appl. 328 (1–2) (2003) 274–286. [16] B. Viswanath, A. Mislove, M. Cha, K. Gummadi, On the evolution of user interaction in facebook, in: Proceedings of the Second ACM Workshop on Online Social Networks, ACM. ; 2009, pp. 37–42. [17] X.F. Wang, X. Li, G.R. Chen, Theory and Application of Complex Networks, Tsinghua University Press, Beijing, 2006. [18] M. Newman, Assortative mixing in networks, Phys. Rev. Lett. 89 (20) (2002) 208701. [19] F. Rodrigues, G. Travieso, V. Boas, Characterization of complex networks: a survey of measurements, Adv. Phys. 56 (1) (2007) 167–242. [20] H.B. Hu, L. Wang, A brief history of power law distributions, Physics 34 (012) (2005) 889–896. [21] R. Albert, A. Barabási, Statistical mechanics of complex networks,, Rev. Mod. Phys. 74 (no. 1) (2002) 47–97. [22] J. Leskovec, J. Kleinberg, C. Faloutsos, Graphs over time: densification laws, shrinking diameters and possible explanations, in: Proceedings of the 11th ACM SIGKDD International Conference on Knowledge Discovery in Data Mining. , 2005, pp. 177–187. [23] S. Asur, S. Parthasarathy, D. Ucar, An event-based framework for characterizing the evolutionary behavior of interaction graphs, ACM Trans. Knowledge Discov. Data (TKDD) 3 (4) (2009) 1–36.
Xueguang Wang is professor of East China University of Political Science and Law. Visiting Academic at University College London (UCL, UK). He received the PhD degree in Computer Networks at East China University of Science and Technology. His main research interests are: computer networks, information security and electronic evidence. He has published research articles in scientific world journal and Applied Mathematics & Information Sciences.
Please cite this article as: X.-G. Wang, A network evolution model based on community structure, Neurocomputing (2015), http://dx. doi.org/10.1016/j.neucom.2015.05.021i