Observing the Evolution of Social Network on Weibo by Sampled Data

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1 Observing the Evolution of Social Network on Weibo by Sampled Data Lu Ma, Gang Lu, Junxia Guo College of Information Science and Technology Beijing University of Chemical Technology Beijing, China Abstract Although there have been many researches on the online social networks (OSNs), observing the evolution of a real OSN is still interesting and instructive for understanding people s behavior in OSNs. In this paper, the actual evolution of the social graph of a real OSN Weibo, is studied by sampled data. The exact timestamp of creating or removing each following relationship cannot be sampled. However, by the created time of the users accounts, the evolution of the social network of Weibo is roughly observed. In this way, it is found that the growing pattern of the network scale shows S-shape. Some other properties of the network, such as network density, the number of connected components, the efficiency of the network, clustering coefficient, degree assortativity, and so on, are also observed. As the network grows, the density of the network keeps reducing and eventually reaches a steady state. The change of the number of connected components indicates the users crowd behavior during the network evolution. Keywords Online social networks; Complex network; Structural evolution; Crowd behavior I. INTRODUCTION There has been a huge rise in the growth of OSNs around the world in recent years, such as Twitter, Facebook, LinkedIn, and so on. OSNs provide a private space for people and tools for interacting with others on the Internet. The OSNs have become the main platform for people to gather information and make friends with like-minded people. Therefore, it is important for researchers who are interested in human behaviors to comprehend the statistics and dynamics of OSNs. Recently, many studies have been carried out on network structure [1-7]. The social structure of Facebook friendships network was investigated, including assortativity coefficients and community structures based on different user characteristics [4]. The growth pattern shows S-shape in recent work by Yang and Chen [8] about the evolution of the discussion network. Hu and Wang [9] also found the S-Shape growth pattern in their research on the evolution of large OSNs. However, in OSNs the evolution curves are smoother than those in the discussion network, especially in the initial stage [9]. In recent work, Gong and Xu [10] observed that the density of the Google+ as a function of time is non-monotone. Facebook became denser over time, which is observed by Backstrom L. et al. [11]. However, the density of the network decreases with the growth of the scale, which is also found in [8]. Most real graphs [12] densify over time, with the number of edges growing super-linearly in the number of nodes. Hu and Wang [9] found that the evolutions of network density, clustering, heterogeneity and modularity exhibit complex, non-monotone feature. In a recent work on Sina microblog, it is found that the microblog group has a shorter diameter of connected graph, average path length and larger average clustering coefficient, which means that the microblog social network is a closely connected group and has small-world characteristic [16]. It is found that the number of microblog user bidirectional friends approximately corresponds with the lognormal distribution in the Sina Weibo [21]. There are many other researches about assortativity [17-20]. In this paper, we focus on Weibo the most popular online social network in China. It is similar to Twitter, and designed especially for Chinese with more features than Twitter. To the best of the authors knowledge, there has been no report about the real observation of the evolution of the social network in Weibo. The main reason is that the whole data set of the large social network with real evolution timestamp of Weibo is hard to obtain. However, a compromised method is introduced here for observing the network evolution roughly. By taking user accounts as the nodes and following relations as directed edges, the creating time of a user account is treated as the time when the node and relevant edges join in the network. In this way, we are able to reveal some evolution features of the social network of Weibo. It is found that the growth pattern of the network scale shows S-shape. And the density of the network decreases continually. We also find that most edges are created between a new user and an old one. It is also found that the average path length and the network diameter are almost constants when the network density declines. The network shows degree disassortativity as well. II. DATA SETS Due to the issues of business and privacy, the whole data set of Weibo cannot be opened to public. However, the open API of Weibo can be used to collect data. By the API, we collected data by BFS from May 30th, 2011 until January 1st, 2012, mainly including information of the users and their following relations. Because the users information and their /16/$31.00 copyright 2016 IEEE ICIS 2016, June 26-29, 2016, Okayama, Japan

2 relations are collected by different threads [15], there are many nodes without their users information in the whole relation network. In order to show the evolution of the network by the created time of the user account, if any user s information is not collected in a relation, that relation is deleted from the whole network. Finally, we ve got a Weibo social network of 3,585,761 users with their information and 14,842,699 following relations between them. The created time of the users account range from August 14th, 2009 to January 6th, Maybe it is worth noticing that Sina Corporation released the private beta of Weibo on 14th, 2009, and our data collecting stopped on January 6th, That indicates our dataset includes the very beginning users and the newest users. It is observed that during the evolution, the network of the dataset is not always connected. Sometimes it has two connected components, while sometimes it has one connected component. Besides, there are some isolated nodes which are not linked to any other node at most time, but finally all the nodes are connected as one network. In the following detailed analysis, only the connected nodes are referred. III. STRUCTURAL EVOLUTION Like Twitter, every registered user of Weibo follows some other users, and is followed by some other users. By taking the users as the node set V, and the relations as the directed edge set E, the social network of Weibo is then abstracted as an directed complex network G(V, E). Because the time when a following relation is created or canceled cannot be obtained, we introduced a compromised way to observe the evolution of the collected social network of Weibo as follows. There is a feature in user information called CreatedAt, which records the time of creating the account by the user. As soon as the account is created, the corresponding node with its following relations between other existing nodes is considered as added to the network. Though we cannot know the exact time stamps of creating or deleting edges, a rough evolution process of the social network of Weibo still can be observed in this way. The private beta of Weibo was released on August 14 th, 2009, and on August 28 th, 2009, Weibo was opened for public registration. In the following detailed analysis, we will use the snapshots of every 7 days starting from August 28 th, 2009 in our dataset. However, Table I lists the numbers of nodes and edges of the network before public registration opened in our dataset. From the table, it can be seen that during the private beta time, the network grew slowly. Once it was opened to the public on August 28 th, the network started growing rapidly. By taking N as the number of nodes and E as the number of edges in the network, Fig. 1 shows the growth of the numbers of connected nodes and edges, and the variation of network density over time. The density of a directed network is defined as d=e/[n(n-1)]. We can find that the density of the network decreases as the network grows. One possible reason for this is that, for most users, they usually follow few person. Therefore, the density declines quickly as more people join in. In Fig. 1, the growth pattern for the network scale, including the number of nodes and edges, shows an S-shape that can be simulated by a logistic function. It was also reported in some other recent works. Fig. 2 shows the increment of the number of nodes and edges at the Tth week for the whole network. Most users and relations appear in the network over a short period of time. It is clear that the number of edges grows much faster than nodes. The reason is that one new node usually brings several new edges. It is clear that each new link can be created between two old users (Old-Old), two new users (New-New), or an old user and a new one (Old-New). As a result, the newly created edges can be classified into three classes as well. Fig. 3 shows the evolution of the proportion of the three kind of edges established in the Tth week. We find that most edges were created between a new user and an old one. This may be because Weibo will recommend existing users to new users to follow. New users usually select existing users to follow by themselves as well. It also indicates that the possibility of new users knowing each other to create bidirectional edges is very small. Due to the limitation of the data set, we can t observe the evolution of the Old-Old edges. Fig. 4 illustrates the number of edges versus the number of nodes in log-log scale. This plot shows that the number of edges grows super-linearly with the number of nodes. When the two edges in different direction between two nodes are combined into one Bidirectional Edge (BE), the rest are Unidirectional Edges (UE). We also evaluate the proportion of BEs and UEs in the network, as Fig. 5 illustrates. UEs are much more than Bes. It may be because most users follow the celebrities but the celebrities don t follow them. TABLE I. THE GROWTH OF THE NETWORK IN THE PERIOD FROM PRIVATE BETA TO PUBLIC REGISTRATION OF WEIBO Date of the snapshot # of total # of connected # of nodes nodes edges 2009/8/ /8/ /8/ /8/ /8/27 1, /8/28 7,715 5,652 45,874 Fig. 1. Evolution of the numbers of nodes, edges, and network density.

3 Fig. 2. The number of new nodes and edges at the Tth week. Fig. 3. Evolution of the proportion of the two kinds of edges. observe the evolution of network efficiency [14] instead. The efficiency of the directed network is defined as 1 1 E =, where d ij = when the node i and j N i j ( N 1) dij belong to two disconnected component. Therefore, 1/d ij =0 when there is no path in the graph between i and j. It can measure how efficiently the network exchanges information. The evolution of the network efficiency over time was shown in Fig. 6. For our dataset, the efficiency of the Weibo decreases along the time and finally it reaches equilibrium. However, just being curious, we also evaluate the average shortest path length (ASPL) and the diameter (D) by taking the distance between unconnected nodes as zero. We would like to call the results as Non-Connected Average Path Length (NC- ASPL) and Non-Connected Diameter (NC-D). The calculation may be different from the original definition of distance between two nodes, but we think maybe the results are interesting. The evolution of the NC-ASPL and NC-D over time compared to the network density d is shown in Fig. 7, together with the number of connected components in the whole network. We can see that NC-ASPL and NC-D are almost constants while the network density declines. But there is no obvious evidence telling these two metrics are analytically connected to each other. We find that NC-ASPL between pairs of users is about 3.8. The shortest path lengths between individuals, the so-called six degree of separation found by Stanley Milgram s experiments [13] investigating the social network of the United States, are here seen in the Weibo on a global scale. Fig. 8 shows the evolution of the clustering coefficients denoted by CC for the network. It shows significantly positive relation with the network density. Fig. 9 shows the evolution of the degree assortativity. The network shows disassortative. The asssortativity value is always less than 0 and never grows positive, suggesting that users usually connect others that with the dissimilar degrees. Fig. 4. Number of edges versus number of nodes, in log-log scales. When we try to evaluate the average path length and the diameter of the network, it is found that the network is not always connected. Sometimes there is only one connected component, while sometimes there are two in the whole network. Even at the end of the evolution, there are still two connected components. That indicates an important feature of OSN: the network evolves with every node going its own way in parallel during the whole evolution process. As a result, we Fig. 5. The evolution of BE and UE.

4 Fig. 6. The evolution of the network efficiency over time. Fig. 9. Evolution of the degree assortativity of Weibo. Fig. 7. The evolution of the NC-ASPL and NC-D over time compared to the network density d, together with the number of connected components (NCC) in the whole network. IV. SUMMARY AND DISCUSSION Because the exact creating time of the edges cannot be obtained by the crawled data from Weibo, the creating time of the users accounts is used as the timestamps of the network evolution. In this way, we study the structural evolution of user relation network of Weibo. We find that the growth scale of the network shows S-shape, which may provide an exemplification for Bass diffusion model [22-24]. The density of the network decreases continually. It is also found that most edges were created between a new user and an old one. For the network is disconnected, we evaluate the network efficiency, which can be used to measure how efficiently information is exchanged in the network. It decreases along the time and finally reaches equilibrium. Basing on our self-defined NC-APSL and NC-D, the average path length and diameter of the network are also evaluated. It is found that the NC-APSL and the NC-D are almost constants when the network density declines. The network shows degree disassortativity as well. The change of the number of connected components in the whole network reflects the people s crowd behavior in parallel during the whole evolution process of the network. That is an interesting feature of OSN, whose evolution is driven by the users behaviors. Maybe we can draw inspiration from the feature for modeling OSN. It will also be interesting to compare the evolution processes of OSNs sampled by different sampling methods. Fig. 8. The evolution of clustering coefficients CC for the largest connected subgraph over time in comparison with the network density d. REFERENCES [1] Y.Y. Ahn, S. Han, H. Kwak, S. Moon, H. Jeong. Analysis of topological characteristics of huge online social networking services. Proc. 16th WWW, ACM Press, New York (2007), pp [2] A. Mislove, M. Marcon, K.P. Gummadi, P. Druschel, B. Bhattacharjee. Proc. 7th IMC. ACM Press, New York (2007), pp [3] F. Fu, L.H. Liu, L. Wang. Empirical analysis of online social networks in the age of Web 2.0. Physica A, 387 (2008), pp [4] A.L. Traud, P.J. Mucha, M.A. Porter. Social structure of Facebook networks. Physica A, 391 (2012), pp [5] X. Xiong, W. Cao, X. Zhou, Y. Hu. Research on the feature model of the formation and evolution of social networks. J. Sichuan Univ. (Eng. Sci. Ed.), 44 (2012), pp

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