Minimizing Seed Set Selection with Probabilistic Coverage Guarantee in a Social Network

A topic propagating in a social network reaches its tipping point if the number of users discussing it in the network exceeds a critical threshold such that a wide cascade on the topic is likely to occur. In this paper, we consider the task of selecting initial seed users of a topic with minimum size so that with a guaranteed probability the number of users discussing the topic would reach a given threshold. We formulate the task as an optimization problem called seed minimization with probabilistic coverage guarantee (SM-PCG). This problem departs from the previous studies on social influence maximization or seed minimization because it considers influence coverage with probabilistic guarantees instead of guarantees on expected influence coverage. We show that the problem is not submodular, and thus is harder than previously studied problems based on submodular function optimization. We provide an approximation algorithm and show that it approximates the optimal solution with both a multiplicative ratio and an additive error. The multiplicative ratio is tight while the additive error would be small if influence coverage distributions of certain seed sets are well concentrated. For one-way bipartite graphs we analytically prove the concentration condition and obtain an approximation algorithm with an $O(\log n)$ multiplicative ratio and an $O(\sqrt{n})$ additive error, where $n$ is the total number of nodes in the social graph. Moreover, we empirically verify the concentration condition in real-world networks and experimentally demonstrate the effectiveness of our proposed algorithm comparing to commonly adopted benchmark algorithms.Comment: Conference version will appear in KDD 201

Beyond Worst-Case (In)approximability of Nonsubmodular Influence Maximization

We consider the problem of maximizing the spread of influence in a social network by choosing a fixed number of initial seeds, formally referred to as the influence maximization problem. It admits a $(1-1/e)$-factor approximation algorithm if the influence function is submodular. Otherwise, in the worst case, the problem is NP-hard to approximate to within a factor of $N^{1-\varepsilon}$. This paper studies whether this worst-case hardness result can be circumvented by making assumptions about either the underlying network topology or the cascade model. All of our assumptions are motivated by many real life social network cascades. First, we present strong inapproximability results for a very restricted class of networks called the (stochastic) hierarchical blockmodel, a special case of the well-studied (stochastic) blockmodel in which relationships between blocks admit a tree structure. We also provide a dynamic-program based polynomial time algorithm which optimally computes a directed variant of the influence maximization problem on hierarchical blockmodel networks. Our algorithm indicates that the inapproximability result is due to the bidirectionality of influence between agent-blocks. Second, we present strong inapproximability results for a class of influence functions that are "almost" submodular, called 2-quasi-submodular. Our inapproximability results hold even for any 2-quasi-submodular $f$ fixed in advance. This result also indicates that the "threshold" between submodularity and nonsubmodularity is sharp, regarding the approximability of influence maximization.Comment: 53 pages, 20 figures; Conference short version - WINE 2017: The 13th Conference on Web and Internet Economics; Journal full version - ACM: Transactions on Computation Theory, 201

Budget Feasible Mechanisms for Experimental Design

In the classical experimental design setting, an experimenter E has access to a population of $n$ potential experiment subjects $i\in \{1,...,n\}$, each associated with a vector of features $x_i\in R^d$. Conducting an experiment with subject $i$ reveals an unknown value $y_i\in R$ to E. E typically assumes some hypothetical relationship between $x_i$'s and $y_i$'s, e.g., $y_i \approx \beta x_i$, and estimates $\beta$ from experiments, e.g., through linear regression. As a proxy for various practical constraints, E may select only a subset of subjects on which to conduct the experiment. We initiate the study of budgeted mechanisms for experimental design. In this setting, E has a budget $B$. Each subject $i$ declares an associated cost $c_i >0$ to be part of the experiment, and must be paid at least her cost. In particular, the Experimental Design Problem (EDP) is to find a set $S$ of subjects for the experiment that maximizes V(S) = \log\det(I_d+\sum_{i\in S}x_i\T{x_i}) under the constraint $\sum_{i\in S}c_i\leq B$; our objective function corresponds to the information gain in parameter $\beta$ that is learned through linear regression methods, and is related to the so-called $D$-optimality criterion. Further, the subjects are strategic and may lie about their costs. We present a deterministic, polynomial time, budget feasible mechanism scheme, that is approximately truthful and yields a constant factor approximation to EDP. In particular, for any small $\delta > 0$ and $\epsilon > 0$, we can construct a (12.98, $\epsilon$)-approximate mechanism that is $\delta$-truthful and runs in polynomial time in both $n$ and $\log\log\frac{B}{\epsilon\delta}$. We also establish that no truthful, budget-feasible algorithms is possible within a factor 2 approximation, and show how to generalize our approach to a wide class of learning problems, beyond linear regression

How to Influence People with Partial Incentives

We study the power of fractional allocations of resources to maximize influence in a network. This work extends in a natural way the well-studied model by Kempe, Kleinberg, and Tardos (2003), where a designer selects a (small) seed set of nodes in a social network to influence directly, this influence cascades when other nodes reach certain thresholds of neighbor influence, and the goal is to maximize the final number of influenced nodes. Despite extensive study from both practical and theoretical viewpoints, this model limits the designer to a binary choice for each node, with no way to apply intermediate levels of influence. This model captures some settings precisely, e.g. exposure to an idea or pathogen, but it fails to capture very relevant concerns in others, for example, a manufacturer promoting a new product by distributing five "20% off" coupons instead of giving away one free product. While fractional versions of problems tend to be easier to solve than integral versions, for influence maximization, we show that the two versions have essentially the same computational complexity. On the other hand, the two versions can have vastly different solutions: the added flexibility of fractional allocation can lead to significantly improved influence. Our main theoretical contribution is to show how to adapt the major positive results from the integral case to the fractional case. Specifically, Mossel and Roch (2006) used the submodularity of influence to obtain their integral results; we introduce a new notion of continuous submodularity, and use this to obtain matching fractional results. We conclude that we can achieve the same greedy $(1-1/e-\epsilon)$-approximation for the fractional case as the integral case. In practice, we find that the fractional model performs substantially better than the integral model, according to simulations on real-world social network data

Coreness of Cooperative Games with Truncated Submodular Profit Functions

Coreness represents solution concepts related to core in cooperative games, which captures the stability of players. Motivated by the scale effect in social networks, economics and other scenario, we study the coreness of cooperative game with truncated submodular profit functions. Specifically, the profit function $f(\cdot)$ is defined by a truncation of a submodular function $\sigma(\cdot)$: $f(\cdot)=\sigma(\cdot)$ if $\sigma(\cdot)\geq\eta$ and $f(\cdot)=0$ otherwise, where $\eta$ is a given threshold. In this paper, we study the core and three core-related concepts of truncated submodular profit cooperative game. We first prove that whether core is empty can be decided in polynomial time and an allocation in core also can be found in polynomial time when core is not empty. When core is empty, we show hardness results and approximation algorithms for computing other core-related concepts including relative least-core value, absolute least-core value and least average dissatisfaction value

Towards Profit Maximization for Online Social Network Providers

Online Social Networks (OSNs) attract billions of users to share information and communicate where viral marketing has emerged as a new way to promote the sales of products. An OSN provider is often hired by an advertiser to conduct viral marketing campaigns. The OSN provider generates revenue from the commission paid by the advertiser which is determined by the spread of its product information. Meanwhile, to propagate influence, the activities performed by users such as viewing video ads normally induce diffusion cost to the OSN provider. In this paper, we aim to find a seed set to optimize a new profit metric that combines the benefit of influence spread with the cost of influence propagation for the OSN provider. Under many diffusion models, our profit metric is the difference between two submodular functions which is challenging to optimize as it is neither submodular nor monotone. We design a general two-phase framework to select seeds for profit maximization and develop several bounds to measure the quality of the seed set constructed. Experimental results with real OSN datasets show that our approach can achieve high approximation guarantees and significantly outperform the baseline algorithms, including state-of-the-art influence maximization algorithms.Comment: INFOCOM 2018 (Full version), 12 page

Coverage, Matching, and Beyond: New Results on Budgeted Mechanism Design

We study a type of reverse (procurement) auction problems in the presence of budget constraints. The general algorithmic problem is to purchase a set of resources, which come at a cost, so as not to exceed a given budget and at the same time maximize a given valuation function. This framework captures the budgeted version of several well known optimization problems, and when the resources are owned by strategic agents the goal is to design truthful and budget feasible mechanisms, i.e. elicit the true cost of the resources and ensure the payments of the mechanism do not exceed the budget. Budget feasibility introduces more challenges in mechanism design, and we study instantiations of this problem for certain classes of submodular and XOS valuation functions. We first obtain mechanisms with an improved approximation ratio for weighted coverage valuations, a special class of submodular functions that has already attracted attention in previous works. We then provide a general scheme for designing randomized and deterministic polynomial time mechanisms for a class of XOS problems. This class contains problems whose feasible set forms an independence system (a more general structure than matroids), and some representative problems include, among others, finding maximum weighted matchings, maximum weighted matroid members, and maximum weighted 3D-matchings. For most of these problems, only randomized mechanisms with very high approximation ratios were known prior to our results

An Efficient Streaming Algorithm for the Submodular Cover Problem

We initiate the study of the classical Submodular Cover (SC) problem in the data streaming model which we refer to as the Streaming Submodular Cover (SSC). We show that any single pass streaming algorithm using sublinear memory in the size of the stream will fail to provide any non-trivial approximation guarantees for SSC. Hence, we consider a relaxed version of SSC, where we only seek to find a partial cover. We design the first Efficient bicriteria Submodular Cover Streaming (ESC-Streaming) algorithm for this problem, and provide theoretical guarantees for its performance supported by numerical evidence. Our algorithm finds solutions that are competitive with the near-optimal offline greedy algorithm despite requiring only a single pass over the data stream. In our numerical experiments, we evaluate the performance of ESC-Streaming on active set selection and large-scale graph cover problems.Comment: To appear in NIPS'1