In many real-world situations, different and often opposite opinions, innovations, or products are competing with one another for their social influence in a networked society. In this paper, we study competitive influence propagation in social networks under the competitive linear threshold (CLT) model, an extension to the classic linear threshold model. Under the CLT model, we focus on the problem that one entity tries to block the influence propagation of its competing entity as much as possible by strategically selecting a number of seed nodes that could initiate its own influence propagation. We call this problem the influence blocking maximization (IBM) problem. We prove that the objective function of IBM in the CLT model is submodular, and thus a greedy algorithm could achieve 1-1/e approximation ratio. However, the greedy algorithm requires Monte-Carlo simulations of competitive influence propagation, which makes the algorithm not efficient. We design an efficient algorithm CLDAG, which utilizes the properties of the CLT model, to address this issue. We conduct extensive simulations of CLDAG, the greedy algorithm, and other baseline algorithms on real-world and synthetic datasets. Our results show that CLDAG is able to provide best accuracy in par with the greedy algorithm and often better than other algorithms, while it is two orders of magnitude faster than the greedy algorithm.
With the increasing popularity of online social and information networks such as Facebook, Twitter, LinkedIn, etc., many researchers have studied diffusion phenomenon in social networks, which includes the diffusion of news, ideas, innovations, adoption of new products, etc. We generally refer to such diffusions as influence diffusion or propagation. One topic in influence diffusion that has been extensively studied is influence maximization [14,15,16,19,6,5,25,7]. Influence maximization is the problem of selecting a small set of seed nodes in a social network, such that its overall influence coverage is maximized, under certain influence diffusion models. Popular influence diffusion models include the independent cascade (IC) model and the linear thresh-old (LT) model, which was first summarized by Kempe et al. in [14] based on prior research in social network analysis and particle physics. Both IC and LT models are stochastic models characterizing how influence are propagated throughout the network starting from the initial seed nodes.
However, all of the above research works only study the diffusion of a single idea in the social networks. In reality, it is often the case that different and often opposite information, ideas and innovations are competing for their influence in the social networks. Such competing influence diffusion could range from two competing companies engaging in two marketing campaigns trying to grab people’s attentions, or two political candidates of the opposing parties trying to influence their voters, to government authorities trying to inject truth information to fight with rumors spreading in the public, and so on.
Motivated by the above scenarios, several recent studies have looked into competitive influence diffusion and its corresponding influence maximization problems [1,17,21,24,2,3,4]. Most of them propose some extensions to the existing influence diffusion models to incorporate competitive influence diffusion, and then either focus on the influence maximization problem for one of the competing parties, or study the game theoretic aspects of competitive influence diffusion (see Section 2 for more details on these related works). In this paper, we concentrate on the problem of how to block the influence diffusion of an opposing party as much as possible. For example, when there is a negative rumor spreading in the social network about a company, the company may want to react quickly by selecting seed nodes to inject positive opinions about the company to fight against the negative rumor. Similar situations could occur when a political candidate tries to stop a negative rumor about him or her, or when government or public officials try to stop erroneous rumors arXiv:1110.4723v1 [cs.SI] 21 Oct 2011 about public health and safety, terrorist threat, etc. We call the problem of selecting positive seed nodes in a social network to minimize the effect of negative influence diffusion, or to maximize the blocking effect on negative influence, the influence blocking maximization (IBM) problem.
We study the IBM problem under a competitive linear threshold (CLT) model, which we extend naturally from the classic linear threshold model and is similar to a model proposed independently in [2]. We prove that the objective function of IBM under the CLT model is monotone and submodular, which means a standard greedy algorithm can achieve an approximation ratio of 1 -1/eto the optimal solution, where is any positive number. However, the greedy algorithm requires Monte-Carlo simulations of competitive influence diffusion, which becomes very slow for large networks, if we want to keep above small. For example, in our simulation, for a network with 6.4k nodes, the greedy algorithm takes more than 8 hours to finish. This is especially problematic for the IBM problem, since blocking influence diffusion usually requires very swift decisions before the negative influence propagates too far. To address the efficiency issue, we utilize the efficient computation property of the LT model for directed acyclic graphs (DAGs), and design an efficient heuristic CLDAG for the IBM problem under the CLT model. Because of the complex interaction in the competitive influence diffusion under the CLT model, we need a carefully designed dynamic programming method for influence computation in our CLDAG algorithm. To test the efficiency and effectiveness of our CLDAG algorithm, we conduct extensive simulations on three real-world networks as well as synthetic networks. We compare the performance of CLDAG with the greedy algorithm and other heuristic algorithms. Our results show that (a) comparing with the greedy algorithm, our CLDAG algorithm achieves matching influence blocking effect while it runs two orders of magnitude faster; and (b) comparing with other heuristics such as degree-based heuristics, our algorithm consistently performs well and is often better than the other heuristics with a significant margin.
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