Although influence maximization problem has been extensively studied over the past ten years, majority of existing work adopt one of the following models: or . In the zero-feedback model, we have to commit the seed users all at once in advance, this strategy is also known as non-adaptive policy. In the full-feedback model, we select one seed at a time and wait until the diffusion completes, before selecting the next seed. Full-feedback model has better performance but potentially huge delay, zero-feedback model has zero delay but poorer performance since it does not utilize the observation that may be made during the seeding process. To fill the gap between these two models, we propose , which allows us to select a seed at any intermediate stage. We develop a novel @math -greedy policy that, for the first time, achieves a bounded approximation ratio.
The first category is non-adaptive influence maximization: we must find a set of influential customers all at once in advance subject to a budget constraint. first formalized and studied this problem under two diffusion models, namely Independent Cascade model and Linear Threshold model. study influence maximization problem under various extended models. The second category is adaptive influence maximization, which is closely related to adaptive stochastic submodular maximization @cite_3 @cite_1 @cite_0 @cite_2 . Existing studies mainly adopt full-feedback model, assuming that we can observe the full status of the previous cascade before selecting the next seed. We relax this assumption by incorporating partial-feedback and develop a novel @math -greedy adaptive policy that achieves the first bounded approximation ratio under partial-feedback model.
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It has been reported that 40 of consumers will share an email offer with their friend and 28 of consumers will share deals via social media platforms. This motivates us to study the influence maximization discount allocation problem: given a social network and a limited marketing budget, which set of initial users should be selected to receive the discount, and how much should the discounts be worth? Our goal is to maximize the number of customers who finally adopt the target product. We investigate this problem under both non-adaptive and adaptive settings. In the first setting, we have to commit the set of initial users and corresponding discounts all at once in advance. In the latter case, given that an user has been offered a discount, we are able to know immediately her decision on whether or not to accept that discount, therefore, the decision process is performed in a sequential manner based on the feedback from previously selected users. We propose a simple greedy policy with an approximation ratio of (1-1 e) in non-adaptive setting. For the significantly more complex adaptive setting, we propose a series of adaptive policies with bounded approximation ratio in terms of expected utility. A wide range of AI problems, such as sensor placement, active learning, and network influence maximization, require sequentially selecting elements from a large set with the goal of optimizing the utility of the selected subset. Moreover, each element that is picked may provide stochastic feedback, which can be used to make smarter decisions about future selections. Finding efficient policies for this general class of adaptive optimization problems can be extremely hard. However, when the objective function is adaptive monotone and adaptive submodular, a simple greedy policy attains a 1 - 1 e approximation ratio in terms of expected utility. Unfortunately, many practical objective functions are naturally non-monotone; to our knowledge, no existing policy has provable performance guarantees when the assumption of adaptive monotonicity is lifted. We propose the adaptive random greedy policy for maximizing adaptive submodular functions, and prove that it retains the aforementioned 1 - 1 e approximation ratio for functions that are also adaptive monotone, while it additionally provides a 1 e approximation ratio for nonmonotone adaptive submodular functions. We showcase the benefits of adaptivity on three real-world network data sets using two non-monotone functions, representative of two classes of commonly encountered non-monotone objectives. Social networks are commonly used for marketing purposes. For example, free samples of a product can be given to a few influential social network users (or seed nodes), with the hope that they will convince their friends to buy it. One way to formalize this objective is through the problem of influence maximization (or IM), whose goal is to find the best seed nodes to activate under a fixed budget, so that the number of people who get influenced in the end is maximized. Solutions to IM rely on the influence probability that a user influences another one. However, this probability information may be unavailable or incomplete. In this paper, we study IM in the absence of complete information on influence probability. We call this problem Online Influence Maximization (OIM), since we learn influence probabilities at the same time we run influence campaigns. To solve OIM, we propose a multiple-trial approach, where (1) some seed nodes are selected based on existing influence information; (2) an influence campaign is started with these seed nodes; and (3) user feedback is used to update influence information. We adopt Explore-Exploit strategies, which can select seed nodes using either the current influence probability estimation (exploit), or the confidence bound on the estimation (explore). Any existing IM algorithm can be used in this framework. We also develop an incremental algorithm that can significantly reduce the overhead of handling user feedback information. Our experiments show that our solution is more effective than traditional IM methods on the partial information. Propagation of contagion through networks is a fundamental process. It is used to model the spread of information, influence, or a viral infection. Diffusion patterns can be specified by a probabilistic model, such as Independent Cascade (IC), or captured by a set of representative traces. Basic computational problems in the study of diffusion are influence queries (determining the potency of a specified seed set of nodes) and Influence Maximization (identifying the most influential seed set of a given size). Answering each influence query involves many edge traversals, and does not scale when there are many queries on very large graphs. The gold standard for Influence Maximization is the greedy algorithm, which iteratively adds to the seed set a node maximizing the marginal gain in influence. Greedy has a guaranteed approximation ratio of at least (1-1 e) and actually produces a sequence of nodes, with each prefix having approximation guarantee with respect to the same-size optimum. Since Greedy does not scale well beyond a few million edges, for larger inputs one must currently use either heuristics or alternative algorithms designed for a pre-specified small seed set size. We develop a novel sketch-based design for influence computation. Our greedy Sketch-based Influence Maximization (SKIM) algorithm scales to graphs with billions of edges, with one to two orders of magnitude speedup over the best greedy methods. It still has a guaranteed approximation ratio, and in practice its quality nearly matches that of exact greedy. We also present influence oracles, which use linear-time preprocessing to generate a small sketch for each node, allowing the influence of any seed set to be quickly answered from the sketches of its nodes.