Social influence occurs when one's opinions, emotions, or behaviors are affected by others, intentionally or unintentionally. In this article, we survey recent research progress on social influence analysis. In particular, we first give a brief overview of related background knowledge, and then discuss what is social influence. We try to answer this question in terms of homophily and the process of influence and selection. After that, we focus on describing computational models for social influence including models for influence probability learning and influence diffusion. Finally, we discuss potential applications of social influence. Preliminaries First we introduce some basic knowledge for social network analysis, including related theories in sociology, fundamental models underlying social networks, and algorithms or measures for quantifying social influence. For social theories, we introduce social balance theory [10], social status theory [21], structural hole theory [6, 22], and two-step information flow theory [19]. For social network models, we introduce the Erdos-Renyi (ER) model 1111, Small-World model [28], and the Barabasi-Albert (BA) model [4]. For algorithms, we review several fundamental problems in graph theory and the corresponding algorithms to solve them including the Ford-Fulkerson algorithm for maximum flow in a flow network [12], the push-relabel maximum flow algorithm for finding k-densest subgraph [13], and the greedy algorithm for the set covering problem [1]. We will also go through standard measures and concepts of social networks and their connection to social influence such as centrality, clustering coefficient, closeness and betweenness. These measures are fundamental concepts about social network analysis, and are also deeply related to the importance or influence of nodes or edges in the networks. Definition and Existential Test As there is no a formal definition for social influence, we discuss its definition in terms of several related concepts such as homophily [20], conformity [8, 27], and selection. We further describe methodologies for verifying the existence of influence in various social networks. The methods include shuffle test [2] and randomization test [24]. We will give real world examples to demonstrate how the social influence behaves in different social networks. For example, Bond et al. [5] conducted a randomized controlled trial by delivering political mobilization messages to 61 million Facebook users. Their results verified the existence of social influence on political voting behavior - when one is aware that their friends have made the political votes, their likelihood to vote will significantly increase. Bakshy et al. [3] also conducted randomized controlled trials to verify the existence of social influence on customer responses to advertising in Facebook. Computational Models We now focus on describing the computational models for social influence analysis, with an emphasis on influence quantification and influence diffusion. In particular, for influence quantification, we introduce several popular methods for learning the influence probability between users. For example, Tang et al. [26] presented a Topical Affinity Propagation (TAP) approach to quantify the topic-level social influence in large networks. Goyal et al. [14] presented a method to learn the influence probabilities by counting the number of correlated social actions. Tan et a. [25] proposed a model to learn and distinguish the effects of influence, correlation, and uses' action dependency. Kutzkov et al. [18] extended influence probability learning to the stream data. They showed that the influence probabilities could be learned with one pass over the streaming data using only O(n log n) space, where n is the number of nodes in a network. For influence diffusion, we start with several state-of-the-art epidemic models such as Susceptible Infectious-Recovered (SIR) [17], Susceptible-Infectious-Susceptible (SIS), and Susceptible-Infectious-Recovered-Susceptible (SIRS). Then, we focus on the two popular influence maximization model including independent cascaded model and linear threshold model. The problem of influence maximization has been formally defined as an algorithmic problem by Domingos and Richardson [9, 23]. Kempe et al. [16] further presented the independent cascaded model and the linear threshold model, and theoretically proved the NP-hardness of the two models. They defined the problem using submodular functions, with which a natural greedy strategy could obtain a solution that is provably within (1 -1/e) of optimal solution. Chen et al. [7] further developed efficient algorithms to approximately solve the influence maximization problem. Finally we will also discuss several extensions of the basic cascaded and linear threshold models, e.g., [15, 30, 29]. Applications Finally we use several real applications as examples to further demonstrate the usefulness of social influence analysis. We empirically study social influence in more than 10 datasets including Twitter, Weibo, Flickr, Gowalla, Coauthor, Mobile, Slash-dot, Enron, Epinions, etc. We will share our experience and interesting findings when studying social influence on these data.
