Given a network represented by a weighted directed graph G, we consider the problem of finding a bounded cost set of nodes S such that the influence spreading from S in G, within a given time bound, is as large as possible. The dynamic that governs the spread of influence is the following: initially only elements in S are influenced; subsequently at each round, the set of influenced elements is augmented by all nodes in the network that have a sufficiently large number of already influenced neighbors. We prove that the problem is NP-hard, even in simple networks like complete graphs and trees. We also derive a series of positive results. We present exact pseudo-polynomial time algorithms for general trees, that become polynomial time in case the trees are unweighted. This last result improves on previously published results. We also design polynomial time algorithms for general weighted paths and cycles, and for unweighted complete graphs.
It is clear that the @math -MIS problem represents an abstraction of the viral marketing scenario if one makes the reasonable assumption that an individual decides to adopt the products if a suitable number of his her friends have adopted the products. Analogously, the @math -MIS problem can describe other diffusion problems arising in sociological, economical, and biological networks (again see @cite_13 ). Therefore, it comes as no surprise that special cases of our problem (or variants thereof) have recently attracted the attention of the algorithmic community. We shall limit ourselves here to discussing the work that is most directly related to ours, and refer the reader to the monographs @cite_35 @cite_13 for an excellent overview of the area. We just mention that our results also seem to be relevant to other areas, like dynamic monopolies @cite_21 @cite_15 for instance.
@cite_35 @cite_15 @cite_21 @cite_13
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We present a randomised distributed maximal independent set (MIS) algorithm for arbitrary graphs of size @math that halts in time @math with probability @math , each message containing @math bit: thus its bit complexity per channel is @math (the bit complexity is the number of bits we need to solve a distributed task, it measures the communication complexity). We assume that the graph is anonymous: unique identities are not available to distinguish the processes; we only assume that each vertex distinguishes between its neighbours by locally known channel names. Furthermore we do not assume that the size (or an upper bound on the size) of the graph is known. This algorithm is optimal (modulo a multiplicative constant) for the bit complexity and improves the best previous randomised distributed MIS algorithms (deduced from the randomised PRAM algorithm due to Luby) for general graphs which is @math per channel (it halts in time @math and the size of each message is @math ). This result is based on a powerful and general technique for converting unrealistic exchanges of messages containing real numbers drawn at random on each vertex of a network into exchanges of bits. Then we consider a natural question: what is the impact of a vertex inclusion in the MIS on distant vertices? We prove that this impact vanishes rapidly as the distance grows for bounded-degree vertices. We provide a counter-example that shows this result does not hold in general. We prove also that these results remain valid for Luby's algorithm presented by Lynch and by Wattenhofer. This question remains open for the variant given by Peleg. We consider the problem faced by a company that wants to use viral marketing to introduce a new product into a market where a competing product is already being introduced. We assume that consumers will use only one of the two products and will influence their friends in their decision of which product to use. We propose two models for the spread of influence of competing technologies through a social network and consider the influence maximization problem from the follower's perspective. In particular we assume the follower has a fixed budget available that can be used to target a subset of consumers and show that, although it is NP-hard to select the most influential subset to target, it is possible to give an efficient algorithm that is within 63 of optimal. Our computational experiments show that by using knowledge of the social network and the set of consumers targeted by the competitor, the follower may in fact capture a majority of the market by targeting a relatively small set of the right consumers. In this paper we consider a fundamental problem in the area of viral marketing, called T ARGET S ET S ELECTION problem. In a a viral marketing setting, social networks are modeled by graphs with potential customers of a new product as vertices and friend relationships as edges, where each vertex @math is assigned a threshold value @math . The thresholds represent the different latent tendencies of customers (vertices) to buy the new product when their friend (neighbors) do. Consider a repetitive process on social network @math where each vertex @math is associated with two states, active and inactive, which indicate whether @math is persuaded into buying the new product. Suppose we are given a target set @math . Initially, all vertices in @math are inactive. At time step 0, we choose all vertices in @math to become active. Then, at every time step @math , all vertices that were active in time step @math remain active, and we activate any vertex @math if at least @math of its neighbors were active at time step @math . The activation process terminates when no more vertices can get activated. We are interested in the following optimization problem, called T ARGET S ET S ELECTION : Finding a target set @math of smallest possible size that activates all vertices of @math . There is an important and well-studied threshold called strict majority threshold, where for every vertex @math in @math we have @math and @math is the degree of @math in @math . In this paper, we consider the T ARGET S ET S ELECTION problem under strict majority thresholds and focus on three popular regular network structures: cycle permutation graphs, generalized Petersen graphs and torus cordalis. The target of viral marketing on the platform of popular online social networks is to rapidly propagate marketing information at lower cost and increase sales, in which a key problem is how to precisely discover the most influential users in the process of information diffusion. A novel method is proposed in this paper for helping companies to identify such users as seeds to maximize information diffusion in the viral marketing. Firstly, the user trust network oriented to viral marketing and users’ combined interest degree in the network including isolated users are extensively defined. Next, we construct a model considering the time factor to simulate the process of information diffusion in viral marketing and propose a dynamic algorithm description. Finally, experiments are conducted with a real dataset extracted from the famous SNS website Epinions. The experimental results indicate that the proposed algorithm has better scalability and is less time-consuming. Compared with the classical model, the proposed algorithm achieved a better performance than does the classical method on the two aspects of network coverage rate and time-consumption in our four sub-datasets.