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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We propose a new influence model for allocating budgets to advertising channels. Our model captures customer's sensitivity to advertisements as a threshold behavior; a customer is expected to be influenced if the influence he receives exceeds his threshold. Over the threshold model, we discuss two optimization problems. The first one is the budget-constrained influence maximization. We propose two greedy algorithms based on different strategies, and analyze the performance when the influence is submodular. We then introduce a new characteristic to measure the cost-effectiveness of a marketing campaign, that is, the proportion of the resulting influence to the cost spent. We design an almost linear-time approximation algorithm to maximize the cost-effectiveness. Furthermore, we design a better-approximation algorithm based on linear programming for a special case. We conduct thorough experiments to confirm that our algorithms outperform baseline algorithms. We consider the budget allocation problem over bipartite influence model proposed by (, 2012). This problem can be viewed as the well-known influence maximization problem with budget constraints. We first show that this problem and its much more general form fall into a general setting; namely the monotone submodular function maximization over integer lattice subject to a knapsack constraint. Our framework includes 's model, even with a competitor and with cost. We then give a (1 - 1 e)-approximation algorithm for this more general problem. Furthermore, when influence probabilities are nonincreasing, we obtain a faster (1 - 1 e)-approximation algorithm, which runs essentially in linear time in the number of nodes. This allows us to implement our algorithm up to almost 10M edges (indeed, our experiments tell us that we can implement our algorithm up to 1 billion edges. It would approximately take us only 500 seconds.). In the well-studied Influence Maximization problem, the goal is to identify a set of k nodes in a social network whose joint influence on the network is maximized. A large body of recent work has justified research on Influence Maximization models and algorithms with their potential to create societ al or economic value. However, in order to live up to this potential, the algorithms must be robust to large amounts of noise, for they require quantitative estimates of the influence, which individuals exert on each other; ground truth for such quantities is inaccessible, and even decent estimates are very difficult to obtain. We begin to address this concern formally. First, we exhibit simple inputs on which even very small estimation errors may mislead every algorithm into highly suboptimal solutions. Motivated by this observation, we propose the Perturbation Interval model as a framework to characterize the stability of Influence Maximization against noise in the inferred diffusion network. Analyzing the susceptibility of specific instances to estimation errors leads to a clean algorithmic question, which we term the Influence Difference Maximization problem. However, the objective function of Influence Difference Maximization is NP-hard to approximate within a factor of O(n1−e) for any e > 0. Given the infeasibility of diagnosing instability algorithmically, we focus on finding influential users robustly across multiple diffusion settings. We define a Robust Influence Maximization framework wherein an algorithm is presented with a set of influence functions. The algorithm’s goal is to identify a set of k nodes who are simultaneously influential for all influence functions, compared to the (function-specific) optimum solutions. We show strong approximation hardness results for this problem unless the algorithm gets to select at least a logarithmic factor more seeds than the optimum solution. However, when enough extra seeds may be selected, we show that techniques of can be used to approximate the optimum robust influence to within a factor of 1−1 e. We evaluate this bicriteria approximation algorithm against natural heuristics on several real-world datasets. Our experiments indicate that the worst-case hardness does not necessarily translate into bad performance on real-world datasets; all algorithms perform fairly well. In this paper, we address the important issue of uncertainty in the edge influence probability estimates for the well studied influence maximization problem --- the task of finding k seed nodes in a social network to maximize the influence spread. We propose the problem of robust influence maximization, which maximizes the worst-case ratio between the influence spread of the chosen seed set and the optimal seed set, given the uncertainty of the parameter input. We design an algorithm that solves this problem with a solution-dependent bound. We further study uniform sampling and adaptive sampling methods to effectively reduce the uncertainty on parameters and improve the robustness of the influence maximization task. Our empirical results show that parameter uncertainty may greatly affect influence maximization performance and prior studies that learned influence probabilities could lead to poor performance in robust influence maximization due to relatively large uncertainty in parameter estimates, and information cascade based adaptive sampling method may be an effective way to improve the robustness of influence maximization. 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. We study the online influence maximization problem in social networks under the independent cascade model. Specifically, we aim to learn the set of "best influencers" in a social network online while repeatedly interacting with it. We address the challenges of (i) combinatorial action space, since the number of feasible influencer sets grows exponentially with the maximum number of influencers, and (ii) limited feedback, since only the influenced portion of the network is observed. Under a stochastic semi-bandit feedback, we propose and analyze IMLinUCB, a computationally efficient UCB-based algorithm. Our bounds on the cumulative regret are polynomial in all quantities of interest, achieve near-optimal dependence on the number of interactions and reflect the topology of the network and the activation probabilities of its edges, thereby giving insights on the problem complexity. To the best of our knowledge, these are the first such results. Our experiments show that in several representative graph topologies, the regret of IMLinUCB scales as suggested by our upper bounds. IMLinUCB permits linear generalization and thus is both statistically and computationally suitable for large-scale problems. Our experiments also show that IMLinUCB with linear generalization can lead to low regret in real-world online influence maximization. We consider various stochastic models that incorporate the notion of risk-averseness into the standard 2-stage recourse model, and develop novel techniques for solving the algorithmic problems arising in these models. A key notable feature of our work that distinguishes it from work in some other related models, such as the (standard) budget model and the (demand-) robust model, is that we obtain results in the black-box setting, that is, where one is given only sampling access to the underlying distribution. Our first model, which we call the risk-averse budget model, incorporates the notion of risk-averseness via a probabilistic constraint that restricts the probability (according to the underlying distribution) with which the second-stage cost may exceed a given budget B to at most a given input threshold . We also a consider a closely-related model that we call the risk-averse robust model, where we seek to minimize the first-stage cost and the (1- )-quantile of the second-stage cost. We obtain approximation algorithms for a variety of combinatorial optimization problems including the set cover, vertex cover, multicut on trees, min cut, and facility location problems, in the risk-averse budget and robust models with black-box distributions. We obtain near-optimal solutions that preserve the budget approximately and incur a small blow-up of the probability threshold (both of which are unavoidable). To the best of our knowledge, these are the first approximation results for problems involving probabilistic constraints and black-box distributions. A major component of our results is a fully polynomial approximation scheme for solving the LP-relaxation of the risk-averse problem. We show that the optimal decision policy for several types of Bayesian sequential detection problems has a threshold switching curve structure on the space of posterior distributions. This is established by using lattice programming and stochastic orders in a partially observed Markov decision process (POMDP) framework. A stochastic gradient algorithm is presented to estimate the optimal linear approximation to this threshold curve. We illustrate these results by first considering quickest time detection with phase-type distributed change time and a variance stopping penalty. Then it is proved that the threshold switching curve also arises in several other Bayesian decision problems such as quickest transient detection, exponential delay (risk-sensitive) penalties, stopping time problems in social learning, and multi-agent scheduling in a changing world. Using Blackwell dominance, it is shown that for dynamic decision making problems, the optimal decision policy is lower bounded by a myopic policy. Finally, it is shown how the achievable cost of the optimal decision policy varies with change time distribution by imposing a partial order on transition matrices. 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. 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. We study the problem of maximizing the expected spread of an innovation or behavior within a social network, in the presence of “word-of-mouth” referral. Our work builds on the observation that individuals’ decisions to purchase a product or adopt an innovation are strongly influenced by recommendations from their friends and acquaintances. Understanding and leveraging this influence may thus lead to a much larger spread of the innovation than the traditional view of marketing to individuals in isolation. In this paper, we define a natural and general model of influence propagation that we term the decreasing cascade model, generalizing models used in the sociology and economics communities. In this model, as in related ones, a behavior spreads in a cascading fashion according to a probabilistic rule, beginning with a set of initially “active” nodes. We study the target set selection problem: we wish to choose a set of individuals to target for initial activation, such that the cascade beginning with this active set is as large as possible in expectation. We show that in the decreasing cascade model, a natural greedy algorithm is a 1-1 e-e approximation for selecting a target set of size k. 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. The Adaptive Seeding problem is an algorithmic challenge motivated by influence maximization in social networks: One seeks to select among certain accessible nodes in a network, and then select, adaptively, among neighbors of those nodes as they become accessible in order to maximize a global objective function. More generally, adaptive seeding is a stochastic optimization framework where the choices in the first stage affect the realizations in the second stage, over which we aim to optimize. Our main result is a (1 -- 1 e)2-approximation for the adaptive seeding problem for any monotone submodular function. While adaptive policies are often approximated via non-adaptive policies, our algorithm is based on a novel method we call locally-adaptive policies. These policies combine a non-adaptive global structure, with local adaptive optimizations. This method enables the (1 -- 1 e)2-approximation for general monotone submodular functions and circumvents some of the impossibilities associated with non-adaptive policies. We also introduce a fundamental problem in submodular optimization that may be of independent interest: given a ground set of elements where every element appears with some small probability, find a set of expected size at most k that has the highest expected value over the realization of the elements. We show a surprising result: there are classes of monotone submodular functions (including coverage) that can be approximated almost optimally as the probability vanishes. For general monotone submodular functions we show via a reduction from P lanted -C lique that approximations for this problem are not likely to be obtainable. This optimization problem is an important tool for adaptive seeding via non-adaptive policies, and its hardness motivates the introduction of locally-adaptive policies we use in the main result. The majority of influence maximization (IM) studies focus on targeting influential seeders to trigger substantial information spread in social networks. In this paper, we consider a new and complementary problem of how to further increase the influence spread of given seeders. Our study is motivated by the observation that direct incentives could "boost" users so that they are more likely to be influenced by friends. We study the k-boosting problem which aims to find k users to boost so that the final "boosted" influence spread is maximized. The k-boosting problem is different from the IM problem because boosted users behave differently from seeders: boosted users are initially uninfluenced and we only increase their probability to be influenced. Our work also complements the IM studies because we focus on triggering larger influence spread on the basis of given seeders. Both the NP-hardness of the problem and the non-submodularity of the objective function pose challenges to the k-boosting problem. To tackle the problem, we devise two efficient algorithms with the data-dependent approximation ratio. We conduct extensive experiments using real social networks demonstrating the efficiency and effectiveness of our proposed algorithms. We show that boosting solutions returned by our algorithms achieves boosts of influence that are up to several times higher than those achieved by boosting solutions returned by intuitive baselines, which have no guarantee of solution quality. We also explore the "budget allocation" problem in our experiments. Compared with targeting seeders with all budget, larger influence spread is achieved when we allocation the budget to both seeders and boosted users. This also shows that our study complements the IM studies. Maximization of submodular functions has wide applications in machine learning and artificial intelligence. Adaptive submodular maximization has been traditionally studied under the assumption that the model of the world, the expected gain of choosing an item given previously selected items and their states, is known. In this paper, we study the setting where the expected gain is initially unknown, and it is learned by interacting repeatedly with the optimized function. We propose an efficient algorithm for solving our problem and prove that its expected cumulative regret increases logarithmically with time. Our regret bound captures the inherent property of submodular maximization, earlier mistakes are more costly than later ones. We refer to our approach as Optimistic Adaptive Submodular Maximization (OASM) because it trades off exploration and exploitation based on the optimism in the face of uncertainty principle. We evaluate our method on a preference elicitation problem and show that non-trivial K-step policies can be learned from just a few hundred interactions with the problem. Influence maximization, defined as finding a small subset of nodes that maximizes spread of influence in social networks, is NP-hard under both Linear Threshold (LT) and Independent Cascade (IC) models, where a line of greedy heuristic algorithms have been proposed. The simple greedy algorithm [14] achieves an approximation ratio of 1-1 e. The advanced CELF algorithm [16], by exploiting the sub modular property of the spread function, runs 700 times faster than the simple greedy algorithm on average. However, CELF is still inefficient [4], as the first iteration calls for N times of spread estimations (N is the number of nodes in networks), which is computationally expensive especially for large networks. To this end, in this paper we derive an upper bound function for the spread function. The bound can be used to reduce the number of Monte-Carlo simulation calls in greedy algorithms, especially in the first iteration of initialization. Based on the upper bound, we propose an efficient Upper Bound based Lazy Forward algorithm (UBLF in short), by incorporating the bound into the CELF algorithm. We test and compare our algorithm with prior algorithms on real-world data sets. Experimental results demonstrate that UBLF, compared with CELF, reduces more than 95 Monte-Carlo simulations and achieves at least 2-5 times speed-raising when the seed set is small. Influence maximization (IM), which selects a set of k users (called seeds) to maximize the influence spread over a social network, is a fundamental problem in a wide range of applications such as viral marketing and network monitoring. Existing IM solutions fail to consider the highly dynamic nature of social influence, which results in either poor seed qualities or long processing time when the network evolves. To address this problem, we define a novel IM query named Stream Influence Maximization (SIM) on social streams. Technically, SIM adopts the sliding window model and maintains a set of k seeds with the largest influence value over the most recent social actions. Next, we propose the Influential Checkpoints (IC) framework to facilitate continuous SIM query processing. The IC framework creates a checkpoint for each window shift and ensures an e-approximate solution. To improve its efficiency, we further devise a Sparse Influential Checkpoints (SIC) framework which selectively keeps O(logN β checkpoints for a sliding window of size N and maintains an e(1−β) 2-approximate solution. Experimental results on both real-world and synthetic datasets confirm the effectiveness and efficiency of our proposed frameworks against the state-of-the-art IM approaches. Influence maximization, the fundamental of viral marketing, aims to find top- @math seed nodes maximizing influence spread under certain spreading models. In this paper, we study influence maximization from a game perspective. We propose a Coordination Game model, in which every individuals make their decisions based on the benefit of coordination with their network neighbors, to study information propagation. Our model serves as the generalization of some existing models, such as Majority Vote model and Linear Threshold model. Under the generalized model, we study the hardness of influence maximization and the approximation guarantee of the greedy algorithm. We also combine several strategies to accelerate the algorithm. Experimental results show that after the acceleration, our algorithm significantly outperforms other heuristics, and it is three orders of magnitude faster than the original greedy method. In this paper, we consider adaptive decision-making problems for stochastic state estimation with partial observations. First, we introduce the concept of weak adaptive submodularity, a generalization of adaptive submodularity, which has found great success in solving challenging adaptive state estimation problems. Then, for the problem of active diagnosis, i.e., discrete state estimation via active sensing, we show that an adaptive greedy policy has a near-optimal performance guarantee when the reward function possesses this property. We further show that the reward function for group-based active diagnosis, which arises in applications such as medical diagnosis and state estimation with persistent sensor faults, is also weakly adaptive submodular. Finally, in experiments of state estimation for an aircraft electrical system with persistent sensor faults, we observe that an adaptive greedy policy performs equally well as an exhaustive search. 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.