Identifying social influence in complex networks: A novel conductance eigenvector centrality model
Introduction
With the explosive growth in the variety and size of social networks, social networks have evolved into one of the well-liked channel for business on the Internet. According to the social media statistics, 16 million small businesses are active on Facebook in 2013. About 86 percent of marketers indicate that social media is important for their business, up from 83 percent in 2012 [1]. For examples, Dell uses social media, like Facebook and Twitter, as an important portal for marketing. Through social marketing, Dell extends its brand, delivers services and shares information with their customers [2]. BrowserCam offers Group-Buying deals through social network websites. Customers are encouraged to disseminate product information in online communities and persuade other less-informed customers to reach the required group size [3]. For business, social network is now far more than a tool but became a trend for innovative business strategies.
Influence identification of social peers is one of the fundamental issues to analyze user behaviors in online social networks [4], [5]. The influential peers in social network are the individuals who are catalysts for promoting diffusion of messages, opinions, innovations and products [6]. The “peer influence analysis” conducted by Forrester Research found that 13.4 percent of users on social networks like Facebook and Twitter generate 80 percent of the online posts that influenced others [7]. The behavior modes of the users in a social network are often determined obviously by the network leaders who are characterized by massive connections and central position in the network [8]. Therefore, identifying influential peers who have potential to promote the company through their networks and offering them special treatments can help business enhance efficiency of information diffusion, reduce marketing cost and increase revenues [9], [10], [11].
Generally speaking, so far three kinds of methods, viz. neighbor-based method, path-based method, and random walk method, have been proposed to calculate peer influence according to the network structure. The neighbor-based method is the most naive and convenient one. Various indexes (e.g. degree centrality [12], [13] and semi-local centrality [14], [15]) and calculation strategies (e.g. k-core decomposition [16], [17], [18]) have been proposed in literature. Although popular for simple procedure and low time complexity to handle the large and complex networks, the neighbor-based methods usually get general ranking results and the influence of peers with the same centrality degree or k-core cannot be differentiated. The path-based methods consider all the paths in a social network and rank peers according to their capabilities to control the information flow in the network. With the path-based methods, peers with high control capabilities (measured by eccentricity [19], [20], closeness centrality [21], [22], betweenness centrality [23], [24], and others [25], [26], [27]) are identified to have high influence. But the shortcoming of the path-based methods is that they trade all neighbors of a peer equally when measure the importance of the peer. To relax the limitation of the path-based methods, the random walk methods calculate the importance of a peer based on both the quantity and quality of its neighbors. These methods distribute initial influential values to each peer and design rules to update the influential values iteratively until stable influential values are obtained. The PageRank algorithm [28], [29], [30], LeaderRank algorithm [31], HITS algorithm [32] are typical random walk algorithms with different update mechanism for peer influence.
Within the random walk framework, this paper proposes a conductance eigenvector centrality (CEC) model to measure the peer influence in social networks. The CEC model assumes that influence is distributed as a conductance network in a social network. Numerous paths exist to deliver influence from the originating peer to the target peer and resistance exists along each path. The resistance between two peers depends on the path length between them and longer path has higher resistance. Influence distribution in the network is assumed as a kind of energy diffusion. Consistent with the actual influence distribution environment, the CEC model is constructed from a global perspective and assumes that all of the paths can deliver energy (influence) from the originating peer to the target peer. Therefore, rich conductance options with shorter paths between two peers means we can obtain more residual energy if we diffuse energy from the originating peer to the target peer, and also means higher influence of the originating peer can be distributed to the target peer than others. This paper proposes a method to calculate the conductance matrix among peers in social networks and proves the existence and uniqueness of the conductance matrix.
With the conductance matrix, the CEC model employs CEC value to measure peer influence and calculates the stable conductance network by the random walk framework. The CEC model distributes initial CEC values to peers, calculates CEC values in each iteration, and updates the conductance network until stable conductance network is obtained. The convergence of the proposed random walk mechanism is proved and we propose a simple method to calculate the convergent CEC values. Compared with the algorithm such as Degree centrality algorithm, K-analysis algorithm, Betweenness centrality algorithm and PageRank algorithm, the experiments show that the proposed CEC model obtains more accurate identification results.
The remainder of this paper is organized as follows. Section 2 introduces four benchmark methods to identify social influence. Section 3 presents an example social network to show the invalidity of the benchmark methods. Section 4 provides the details of the CEC model. Numerical experiments under different scenarios are conducted in Section 5 to validate the proposed model. Discussions, conclusions, and future research are presented in Section 6.
Section snippets
Benchmark algorithms
This section introduces the details of four influence identification methods which are chosen as the benchmark algorithms for the proposed CEC model. The Degree centrality algorithm and the k-core decomposition algorithm are classical neighbor-based methods; the Betweenness centrality algorithm is the representative of the path-based methods; and the PageRank algorithm is a well-known random walk method. This section also introduces the SIR model which is employed to calculate the
An example social network
We use Fig. 2 to illustrate the limitation of the existing methods. Table 1 classifies the 19 peers in Fig. 2 into eight groups and the peers in each group are homogenous from the perspective of network structure. For example, peer 2, 3, 4 and 5 are homogeneous because they have the same link structure in the network and the serial number of the four peers can be interchanged within the group.
We now employ the SIR model to quantify the peer influence in Fig. 2. To provide a robust criterion, we
Conductance eigenvector centrality (CEC) model
The wrong identification results of the benchmark algorithms indicate that new methods are desired to identify social influence more accurately. This paper proposes a conductance eigenvector centrality (CEC) model to measure peer influence in social networks. The CEC model assumes the social network in which influence is distributed as a conductance network. Numerous paths exist to diffuse influence and resistance exists in the diffusion paths. The task of the proposed CEC algorithm in the
Analysis of example social network
Fig. 2 introduced an example social network to illustrate the limitation of the benchmark algorithms. This section analyzes the peer influence by the proposed CEC model. Fig. 5 presents the conductance matrix of the peers in Fig. 2. As shown in Fig. 5, the conductance values among peers are unequal. In the PageRank algorithm, the PR value of a page is divided equally among the outbound links on the page. Fig. 5 shows that this does not hold true when identifying peer influence. Fig. 6 presents
Conclusion
Social network is changing the business model and social influence has been one of the most investigated social issues. This paper proposed a conductance eigenvector centrality (CEC) model to measure the peer influence in social networks. By taking all the paths into account to measure peer influence, this paper proposed a method to calculate the conductance matrix of social networks and provided the proof for existence and uniqueness of the proposed method. We also designed a convergent random
Acknowledgments
This work was supported by the Major Program of National Natural Science Foundation of China (71490725), National Key Basic Research Program of China (2013CB329600), National Natural Science Foundation of China (91546114, 71302064, 71371062, 71501057), National Social Science Foundation of China (13CGL017), National Key Technology Support Program (2015BAH26F00), CCF-Tencent Open Fund (CCF-TencentRAGR20140109), the Key Project of Natural Science of Anhui Province Education Department (KJ2015A348
Xujun Li is an Associate Professor at Administrative School of Anhui, Hefei, China. He is currently a PhD student in Hefei University of Technology. His research interests include electronic commerce and social network analysis. He has published papers in journals such as Journal of System Science and Mathematical Science.
References (48)
- et al.
Towards a formal semantics of social influence
Knowl.-Based Syst.
(2014) - et al.
Development of a group recommender application in a social network
Knowl.-Based Syst.
(2014) - et al.
h-Degree as a basic measure in weighted networks
J. Informetr.
(2011) - et al.
Identifying influential nodes in complex networks
Phys. A: Stat. Mech. Appl.
(2012) - et al.
Ranking the spreading influence in complex networks
Phys. A: Stat. Mech. Appl.
(2013) - et al.
A general conceptual framework for characterizing the ego in a network
J. Informetr.
(2015) - et al.
Mixing local and global information for community detection in large networks
J. Comput. Syst. Sci.
(2014) - et al.
Ranking hubs and authorities using matrix functions
Linear Algebra Appl.
(2013) - et al.
Robust stereo matching using adaptive random walk with restart algorithm
Image Vis. Comput.
(2015) - et al.
Identifying influential nodes in complex networks with community structure
Knowl.-Based Syst.
(2013)
Group buying: a new mechanism for selling through social interactions
Manag. Sci.
New product diffusion with influentials and imitators
Mark. Sci.
The spread of behavior in an online social network experiment
Science
Commentary-identifying social influence: a comment on opinion leadership and social contagion in new product diffusion
Mark. Sci.
Opinion leadership and social contagion in new product diffusion
Mark. Sci.
Networks, social influence, and the choice among competing innovations: insights from open source software licenses
Inf. Syst. Res.
View-based discriminative probabilistic modeling for 3d object retrieval and recognition
Image Process. IEEE Trans.
Multimodal graph-based reranking for web image search
Image Process. IEEE Trans.
K-core-based attack to the internet: is it more malicious than degree-based attack?
World Wide Web
A k-shell decomposition method for weighted networks
New J. Phys.
Event driven web video summarization by tag localization and key-shot identification
Multimed. IEEE Trans.
Cited by (0)
Xujun Li is an Associate Professor at Administrative School of Anhui, Hefei, China. He is currently a PhD student in Hefei University of Technology. His research interests include electronic commerce and social network analysis. He has published papers in journals such as Journal of System Science and Mathematical Science.
Yezheng Liu is a professor of Electronic Commerce at Hefei University of Technology. Dr. Liu received his PhD in management science and engineering from Hefei University of Technology. He teaches electronic commerce, decision sciences, and information systems. His main research interests include data mining and its application in electronic commerce, decision support systems, and optimization models. He has published papers in journals such as Marketing Science, Decision Support Systems, International Journal of Production Economics, Knowledge-Based Systems, International Journal of Information Technology and Decision Making, and Expert Systems with Applications.
Yuanchun Jiang is an Associate Professor at School of Management, Hefei University of Technology. He received his PhD in management science and engineering from Hefei University of Technology, Hefei, China. He teaches electronic commerce, business intelligence and business research methods. His research interests include electronic commerce, online marketing and data mining. He has published papers in journals such as Marketing Science, Decision Support Systems, IEEE Transaction on Software Engineering, International Journal of Production Economics, Journal of Systems and Software, Knowledge-Based Systems, International Journal of Information Technology and Decision Making, and Expert Systems with Applications, among others.
Xiao Liu received the master’s degree in management science and engineering from Hefei University of Technology, Hefei, China, 2007, and received the PhD degree in computer science and software engineering from the Faculty of Information and Communication Technologies at Swinburne University of Technology, Melbourne, Australia, 2011. He worked as a research fellow with the Faculty of Information and Communication Technologies at Swinburne University of Technology from 2011 to 2012. He is currently an associate professor at the Software Engineering Institute, East China Normal University, Shanghai, China. He has published papers in journals such as IEEE Transactions on Software Engineering, ACM Transactions on Software Engineering and Methodology and IEEE Transactions on Parallel and Distributed Systems. He is a member of the IEEE.