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.
Independent cascade model and linear threshold model are two extensively studied influence diffusions models originally summarized by @cite_21 , based on earlier works of @cite_16 @cite_24 @cite_6 . prove that the generalized versions of these two models are equivalent @cite_21 . Based on the IC and LT model, Kempe et.al @cite_21 @cite_15 propose a greedy algorithm to solve the influence maximization problem (brought about by Richardson @cite_8 ) to maximize the spreading of a single piece of ideas, innovations, etc. under these two models. Many follow-up studies propose alternative heuristics and try to solve the influence maximization problem more efficiently @cite_14 @cite_12 @cite_13 @cite_22 @cite_2 @cite_18 . In terms of efficient algorithm design, our work follows the idea in @cite_22 @cite_2 of finding efficient local graph structures to speed up the computation. In particular, our CLDAG algorithm is similar to the LDAG algorithm of @cite_2 , which is also based on the DAG structure, but our CLDAG algorithm is novel in dealing with competitive influence diffusion using the dynamic programming method.
@cite_18 @cite_14 @cite_22 @cite_8 @cite_21 @cite_6 @cite_24 @cite_2 @cite_15 @cite_16 @cite_13 @cite_12
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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. Influence maximization is the problem of finding a small set of most influential nodes in a social network so that their aggregated influence in the network is maximized. In this paper, we study influence maximization in the linear threshold model, one of the important models formalizing the behavior of influence propagation in social networks. We first show that computing exact influence in general networks in the linear threshold model is #P-hard, which closes an open problem left in the seminal work on influence maximization by Kempe, Kleinberg, and Tardos, 2003. As a contrast, we show that computing influence in directed a cyclic graphs (DAGs) can be done in time linear to the size of the graphs. Based on the fast computation in DAGs, we propose the first scalable influence maximization algorithm tailored for the linear threshold model. We conduct extensive simulations to show that our algorithm is scalable to networks with millions of nodes and edges, is orders of magnitude faster than the greedy approximation algorithm proposed by and its optimized versions, and performs consistently among the best algorithms while other heuristic algorithms not design specifically for the linear threshold model have unstable performances on different real-world networks. Given a social network G and a constant @math , the influence maximization problem asks for k nodes in G that (directly and indirectly) influence the largest number of nodes under a pre-defined diffusion model. This problem finds important applications in viral marketing, and has been extensively studied in the literature. Existing algorithms for influence maximization, however, either trade approximation guarantees for practical efficiency, or vice versa. In particular, among the algorithms that achieve constant factor approximations under the prominent independent cascade (IC) model or linear threshold (LT) model, none can handle a million-node graph without incurring prohibitive overheads. This paper presents TIM, an algorithm that aims to bridge the theory and practice in influence maximization. On the theory side, we show that TIM runs in O((k+ l) (n+m) log n e2) expected time and returns a (1-1 e-e)-approximate solution with at least 1 - n-l probability. The time complexity of TIM is near-optimal under the IC model, as it is only a log n factor larger than the Ω(m + n) lower-bound established in previous work (for fixed k, l, and e). Moreover, TIM supports the triggering model, which is a general diffusion model that includes both IC and LT as special cases. On the practice side, TIM incorporates novel heuristics that significantly improve its empirical efficiency without compromising its asymptotic performance. We experimentally evaluate TIM with the largest datasets ever tested in the literature, and show that it outperforms the state-of-the-art solutions (with approximation guarantees) by up to four orders of magnitude in terms of running time. In particular, when k = 50, e = 0.2, and l = 1, TIM requires less than one hour on a commodity machine to process a network with 41.6 million nodes and 1.4 billion edges. This demonstrates that influence maximization algorithms can be made practical while still offering strong theoretical guarantees. The adoption of everyday decisions in public affairs, fashion, movie-going, and consumer behavior is now thoroughly believed to migrate in a population through an influential network. The same diffusion process when being imitated by intention is called viral marketing. This process can be modeled by a (directed) graph G=(V,E) with a threshold t(v) for every vertex v?V, where v becomes active once at least t(v) of its (in-)neighbors are already active. A Perfect Target Set is a set of vertices whose activation will eventually activate the entire graph, and the Perfect Target Set Selection Problem (PTSS) asks for the minimum such initial set. It is known (Chen (2008) 6) that PTSS is hard to approximate, even for some special cases such as bounded-degree graphs, or majority thresholds.We propose a combinatorial model for this dynamic activation process, and use it to represent PTSS and its variants by linear integer programs. This allows one to use standard integer programming solvers for solving small-size PTSS instances. We also show combinatorial lower and upper bounds on the size of the minimum Perfect Target Set. Our upper bound implies that there are always Perfect Target Sets of size at most |V| 2 and 2|V| 3 under majority and strict majority thresholds, respectively, both in directed and undirected graphs. This improves the bounds of 0.727|V| and 0.7732|V| found recently by Chang and Lyuu (2010) 5 for majority and strict majority thresholds in directed graphs, and matches their bound under majority thresholds in undirected graphs. Furthermore, our proof is much simpler, and we observe that some of these bounds are tight. One interesting and perhaps surprising implication of our lower bound for undirected graphs, is that it is easy to get a constant factor approximation for PTSS for “relatively balanced” graphs (e.g., bounded-degree graphs, nearly regular graphs) with a “more than majority” threshold (that is, t(v)=??deg(v), for every v?V and some constant ?>1 2), whereas no polylogarithmic approximation exists for “more than majority” graphs.