Modeling Dynamic Evolution of Online Friendship Network

Commun. Theor. Phys. 58 (2012) 599 603 Vol. 58, No. 4, October 15, 2012 Modeling Dynamic Evolution of Online Friendship Network WU Lian-Ren ( ) 1,2, and YAN Qiang ( Ö) 1 1 School of Economics and Management, Beijing University of Posts and Telecommunications, Beijing 100876, China 2 Department of Electrical Engineering, Arizona State University, Tempe AZ, USA (Received May 2, 2012; revised manuscript received July 2, 2012) Abstract In this paper, we study the dynamic evolution of friendship network in SNS (Social Networking Site). Our analysis suggests that an individual joining a community depends not only on the number of friends he or she has within the community, but also on the friendship network generated by those friends. In addition, we propose a model which is based on two processes: first, connecting nearest neighbors; second, strength driven attachment mechanism. The model reflects two facts: first, in the social network it is a universal phenomenon that two nodes are connected when they have at least one common neighbor; second, new nodes connect more likely to nodes which have larger weights and interactions, a phenomenon called strength driven attachment (also called weight driven attachment). From the simulation results, we find that degree distribution P(k), strength distribution P(s), and degree-strength correlation are all consistent with empirical data. PACS numbers: 89.75.Hc, 87.23.Kg, 05.50.+q Key words: friendship network, common neighbor, CNN, strength driven 1 Introduction The generation of communities and their development over time are the central research issues in the social sciences [1 4] political movements, professional organizations, and religious denominations all provide fundamental examples of such communities. In the era of internet, on-line groups are becoming increasingly prominent due to the growth of community and social networking sites such as Facebook and MySpace. However, the difficulty of collecting and analyzing large-scale timestamp data on social networks has posed some open questions about the evolution of social networks. What factors influence new individuals joining in communities? And what factors determine the growth of communities? Understanding the structure and dynamics of communities is a goal of network analysis. However, the analysis of dynamic communities is still in its infancy. Fortunato insisted that studies in this direction have been mostly hindered by the fact that the problem of graph clustering is already controversial on single graph realizations, so it is understandable that most efforts still concentrate on the static version of the problem. [4] Recently, the rapid development of data mining technology, has made collection of large-scale timestamp data become possible. So it has become possible to investigate how communities form, evolve, and die. Leskovec et al. presented a detailed study of network evolution by analyzing four large online social networks with temporal information about node and edge arrivals. [5] They have also investigated a wide variety of network formation strategies, and shown that edge locality plays a critical role in evolution of networks. Kumar et al. studied the evolution of large online social networks and presented a simple model of network growth. [6] Palla et al. performed a systematic analysis of dynamic communities. [7] They studied two social systems: a graph of phone calls between customers of a mobile phone company in a year s time and a collaboration network between scientists. Backstrom et al. have carried out an analysis of group dynamics in the free online community of LiveJournal and in a co-authorship network of computer scientists. [8] They have found that the probability that an individual joins a community grows with the number of friends/co-authors who are already in the community and with their degree of interconnectedness. In order to investigate the communities generate mechanisms, some models have been proposed. Such as DEB model, which proposed by Davidsen, Ebel, and Bornholdt. [9] The basic assumption of their model is that the evolution of social connections is mainly determined by the creation of new relations between pairs of individuals with a common friend. Vazquez proposed CNN (Connecting Nearest Neighbors) model, suggesting that in social networks it is more probable that two nodes with a common neighbor get connected than two nodes chosen at random. [10] Yuta et al. proposed CNNR model, which based on two processes: connecting nearest neighbors with Supported by Program for New Centurty Excellent Talents in University under Grant No. NCET-11-0597, and the Fundamental Research Funds for the Central Universities under Grant No. 2012RC1002 Corresponding author, E-mail: lianrenw@asu.edu c 2011 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

600 Communications in Theoretical Physics Vol. 58 apparently random linkage. [11] Rudolf et al. presented a detailed analysis of the CNN model by Vazquez. [12] They have also shown that the degree distribution follows a power law, but the scaling exponent can vary with the parameter setting. In this paper, we use the data from Renren Network which is the largest real-name social networking site in China. By observing user s social interaction, we study the formation and evolution of local friendship networks. Furthermore, we propose a model and show that the model can explain some statistical properties including long-tail degree distribution and degree-strength correlation. In Sec. 2, we describe the data and present the basic statistic results. In Sec. 3, we propose a simple model with connecting nearest neighbors and strength-driven attachment, which we refer to as CNNSD. Then we perform numerical simulation of the model and show that it can explain the statistical properties. Section 4 is the conclusion. 2 Social Networking Site and Dataset The Renren Network, formerly known as Xiaonei Network (literally on-campus network ), is a Chinese social networking service that is similar to Facebook. It is popular among college students in China. In April 2011, Renren made an announcement that it had a total of 31 million active monthly users. Renren Network provides an arena on the internet, where millions of people are creating personal pages. After the registration, people communicate with each other by sending messages and thus grow up friendship. Renren has actively collaborated with academic researchers to understand the growth and structure of its user community. Jiang et al. studied the Renren social graph structure, and found it similar to prior studies of Facebook s social graph. [13] There are some kinds of activities on Renren, such as visiting friends homepages, blogging, voting and sharing others blogs. Here, we focus on the activities of visiting interaction (referred to as the interaction). We study 272 students of the School of Economics and Management in the BUPT (Beijing University of Posts and Telecommunications), they are all Renren users. The data set we consider is the interactions record between these 272 Renren users and their 76103 friends in Renren network. In more than two years, 76 375 users received a total of 787,631 visits. For each visitation, the timestamp and visitor were recorded. For each user i, the number of visits who received is marked as strength s i and the number of friends is marked as degree k i. Basic analysis of dataset, we find that the degree distribution and strength distribution of friendship network follow power laws. In addition, we find that the correlation of degree k and strength s is linear in log-log scale (Figs. 1 3). The friendship network comprises 76 375 nodes. Fig. 1 The distribution of degree P(k). Fig. 2 The distribution of strength P(s). Fig. 3 The correlation of degree k and strength s. 3 Model and Simulation CNNSD Model (Connecting nearest neighbors with the strength driven attachment mechanism) In order to investigate the formation and evolution of local friendship network, we propose a CNNSD model. The model reflects two facts: first, in the social network it

No. 4 Communications in Theoretical Physics 601 is a universal phenomenon that two nodes are connected when they have at least one common neighbor; second, new nodes connect more likely to nodes, which have larger weights and interactions, a phenomenon called strength driven attachment. [14 17] The model dynamics starts from an initial seed of N 0 nodes connected by random links, and iteratively performs the following. (i) With probability 1 u, add a new node in the network. Create a link from the new node to a previously existing node i according to the probability distribution s i / j s j. At the same time, create a set of potential edges from the new node to all the neighbors of node i. (ii) With probability u convert one potential edge selected at random into an edge. CNN Model (Connecting Nearest Neighbors model) The CNN model has been inspired by social networks. [10] It is assumed that in such networks two nodes with a common neighbor are connected with greater probability then two randomly chosen nodes. [6,8,10,14] An analytical understanding of the CNN model has been achieved using the notions of potential edges and potential degree. According to Ref. [10], a potential edge is defined as follows: two nodes are connected by a potential edge, if they are not connected by an edge and they have at least one common neighbor. The network dynamics will be defined by the transition rates between the three possible states: disconnected (d), potential edge (p), or an edge (e). For each node i has the set of transition rates: ν x y (i); x, y {d, p, e}. Consider a network with N nodes. We can write the rate equations for the evolution of the number of nodes with degree k and potential degree k. [12] k i N = ν d eˆk i + ν p e k i (ν e d + ν e p )k i, (1) k i N = ν d pˆk i + ν e p k i (ν p d + ν p e )k i, (2) ˆk i = N k i k i, (3) ν x y is the transition rate from state x to state y per unit of N and ˆk i is the number of remaining neighbors, which are not connected by a potential edge or by an edge to node i. If a new nodei is connected to an existing node j then a potential edge is created between i and all neighbors of j. Hence ν d p = ν d e k i, ν p d = ν e d k i. In order to simplify Eqs. (1) (3), Vazquez and Rudolf were introduced several assumptions: [10,12] (i) All processes leading to edge deletion are neglected: ν e p = ν e d = 0. (ii) The transition from a potential edge to an edge has a higher probability of occurrence than the transition from being disconnected to an edge ν p e = µ 1 N, ν d e = µ 0 N 2, where µ 1 > 0 and µ 0 > 0 are constants. Under these approximations the system of Eqs. (1) (3) is reduced to N k i N = µ 0 + µ 1 ki, With solution where N k i N = µ 0k i µ 1 k i. k i (N) = k 0 ( N N i ) β, k i (N) = k 0( N N i ) β, β = µ [ ( 1 1 + 1 + 4 µ ) ] 0. 2 µ 1 The degree distribution of the system is therefore power law with the exponent. [10,12] γ = 1 + 1 β. Simulation Research In the model, we adopt the parameters as the following: u = 0.5, N 0 = 200 and final network size N = 10 5. The results are averaged over 10 independent realizations. From the simulation results (Figs. 4 6), we find that degree distribution P(k), strength distribution P(s), and degree-strength correlation are all consistent with empirical data. Fig. 4 The distribution of degree P(k). Fig. 5 The distribution of strength P(s).

602 Communications in Theoretical Physics Vol. 58 such as weight. [16 17] Using the same way, we fix parameters u = 200, N = 10 5 and let N 0 = 100, N 0 = 200, N 0 = 500, N 0 = 100 respectively. We find that all the exponents did not change. However, clustering coefficient of the network has changed significantly (Fig. 8). Finally, we generate a small friendship network by the proposed model. Network parameters are u = 0.5, N 0 = 10, network size N = 50 (Fig. 9). We can see that most nodes displaying star structure. There have many edges joining nodes of the same community and comparatively few edges joining nodes of different communities. Fig. 6 The correlation of degree k and strength s. Fig. 9 Schematic of friendship networks. Fig. 7 Degree distribution for different values of u, N 0 = 200, N = 10 5, and the results average over 10 independent realizations. Fig. 8 The clustering coefficient depends on the parameter N 0. In order to analyze the influence of parameter u, we fix parameters N 0 = 200, N = 10 5 and let the parameter u = 0.3, u = 0.5, u = 0.7 respectively. We find that only the exponents of degree distribution changed (Fig. 7) and the addition rate u just control the nodes degree. The exponent of strength distribution depends on other factors, 4 Conclusion In recent years, many models have been proposed for the evolution of complex networks. These models are typically advanced in order to reproduce statistical network properties observed in real-world data. In this paper, we investigate the formation and evolution of local friendship networks. In particular, we analyze the structure and evolution of communities, which have different size. We find that the prominent community structure is stars, a common feature of online social networks. This phenomenon indicates that local relation plays a critical role in the evolution of communities. Based on these empirical observations, we propose a model, which is an extension of CNN model. [10] The model reflects two facts: first, in the social network it is a universal phenomenon that two nodes are connected when they have at least one common neighbor; second, new nodes connect more likely to nodes, which have larger weights and interactions, a phenomenon called strength driven attachment. [14 17] The numerical results show that degree distribution P(k), strength distribution P(s), and degree-strength correlation are all consistent with empirical data. Therefore, our present model will demonstrate its applications in the modeling of online social networks.

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