In the adaptive influence maximization problem, we are given a social network and a budget $k$, and we iteratively select $k$ nodes, called seeds, in order to maximize the expected number of nodes that are reached by an influence cascade that they generate according to a stochastic model for influence diffusion. Differently from the non-adaptive influence maximization problem, where all the seeds must be selected beforehand, here nodes are selected sequentially one by one, and the decision on the $i$th seed is based on the observed cascade produced by the first $i-1$ seeds. We focus on the myopic feedback model, in which we can only observe which neighbors of previously selected seeds have been influenced and on the independent cascade model, where each edge is associated with an independent probability of diffusing influence. Previous works showed that the adaptivity gap is at most $4$, which implies that the non-adaptive greedy algorithm guarantees an approximation factor of $\frac{1}{4}\left(1-\frac{1}{e}\right)$ for the adaptive problem. In this paper, we improve the bounds on both the adaptivity gap and on the approximation factor. We directly analyze the approximation factor of the non-adaptive greedy algorithm, without passing through the adaptivity gap, and show that it is at least $\frac{1}{2}\left(1-\frac{1}{e}\right)$. Therefore, the adaptivity gap is at most $\frac{2e}{e-1}\approx 3.164$. To prove these bounds, we introduce a new approach to relate the greedy non-adaptive algorithm to the adaptive optimum. The new approach does not rely on multi-linear extensions or random walks on optimal decision trees, which are commonly used techniques in the field. We believe that it is of independent interest and may be used to analyze other adaptive optimization problems.
We consider the *adaptive influence maximization problem*: given a network and a budget $k$, iteratively select $k$ seeds in the network to maximize the expected number of adopters. In the *full-adoption feedback model*, after selecting each seed, the seed-picker observes all the resulting adoptions. In the *myopic feedback model*, the seed-picker only observes whether each neighbor of the chosen seed adopts. Motivated by the extreme success of greedy-based algorithms/heuristics for influence maximization, we propose the concept of *greedy adaptivity gap*, which compares the performance of the…
In this paper, we study the non-monotone DR-submodular function maximization over integer lattice. Functions over integer lattice have been defined submodular property that is similar to submodularity of set functions. DR-submodular is a further extended submodular concept for functions over the integer lattice, which captures the diminishing return property. Such functions find many applications in machine learning, social networks, wireless networks, etc. The techniques for submodular set function maximization can be applied to DR-submodular function maximization, e.g., the double greedy alg…
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…
Influence maximization, defined as a problem of finding a set of seed nodes to trigger a maximized spread of influence, is crucial to viral marketing on social networks. For practical viral marketing on large scale social networks, it is required that influence maximization algorithms should have both guaranteed accuracy and high scalability. However, existing algorithms suffer a scalability-accuracy dilemma: conventional greedy algorithms guarantee the accuracy with expensive computation, while the scalable heuristic algorithms suffer from unstable accuracy. In this paper, we focus on solving…
We study the adaptive influence maximization problem with myopic feedback under the independent cascade model: one sequentially selects k nodes as seeds one by one from a social network, and each selected seed returns the immediate neighbors it activates as the feedback available for later selections, and the goal is to maximize the expected number of total activated nodes, referred as the influence spread. We show that the adaptivity gap, the ratio between the optimal adaptive influence spread and the optimal non-adaptive influence spread, is at most 4 and at least e/(e-1), and the approximat…
Influence maximization, fundamental for word-of-mouth marketing and viral marketing, aims to find a set of seed nodes maximizing influence spread on social network. Early methods mainly fall into two paradigms with certain benefits and drawbacks: (1)Greedy algorithms, selecting seed nodes one by one, give a guaranteed accuracy relying on the accurate approximation of influence spread with high computational cost; (2)Heuristic algorithms, estimating influence spread using efficient heuristics, have low computational cost but unstable accuracy. We first point out that greedy algorithms are essen…
We study influence maximization on temporal networks. This is a special setting where the influence function is not submodular, and there is no optimality guarantee for solutions achieved via greedy optimization. We perform an exhaustive analysis on both real and synthetic networks. We show that the influence function of randomly sampled sets of seeds often violates the necessary conditions for submodularity. However, when sets of seeds are selected according to the greedy optimization strategy, the influence function behaves effectively as a submodular function. Specifically, violations of th…
We consider the problem of maximizing the spread of influence in a social network by choosing a fixed number of initial seeds --- a central problem in the study of network cascades. The majority of existing work on this problem, formally referred to as the influence maximization problem, is designed for submodular cascades. Despite the empirical evidence that many cascades are non-submodular, little work has been done focusing on non-submodular influence maximization. We propose a new heuristic for solving the influence maximization problem and show via simulations on real-world and synthetic …
Sharif University（谢里夫大学）；Max Planck Institute for Software Systems（马克斯·普朗克软件系统研究所）
The standard greedy algorithm has been recently shown to enjoy approximation guarantees for constrained non-submodular nondecreasing set function maximization. While these recent results allow to better characterize the empirical success of the greedy algorithm, they are only applicable to simple cardinality constraints. In this paper, we study the problem of maximizing a non-submodular nondecreasing set function subject to a general matroid constraint. We first show that the standard greedy algorithm offers an approximation factor of $\frac{0.4 γ^{2}}{\sqrt{γr} + 1}$, where $γ$ is the submodu…
Competitive influence maximization has been studied for several years, and various frameworks have been proposed to model different aspects of information diffusion under the competitive environment. This work presents a new gameboard for two competing parties with some new features representing loyalty in social networks and reflecting the attitude of not completely being loyal to a party when the opponent offers better suggestions. This behavior can be observed in most political occasions where each party tries to attract people by making better suggestions than the opponent and even seeks t…
Modularity maximization using greedy algorithms continues to be a popular approach toward community detection in graphs, even after various better forming algorithms have been proposed. Apart from its clear mechanism and ease of implementation, this approach is persistently popular because, presumably, its risk of algorithmic failure is not well understood. This Rapid Communication provides insight into this issue by estimating the algorithmic performance limit of modularity maximization. This is achieved by counting the number of metastable states under a local update rule. Our results offer …
We consider the influence maximization problem (selecting $k$ seeds in a network maximizing the expected total influence) on undirected graphs under the linear threshold model. On the one hand, we prove that the greedy algorithm always achieves a $(1 - (1 - 1/k)^k + Ω(1/k^3))$-approximation, showing that the greedy algorithm does slightly better on undirected graphs than the generic $(1- (1 - 1/k)^k)$ bound which also applies to directed graphs. On the other hand, we show that substantial improvement on this bound is impossible by presenting an example where the greedy algorithm can obtain at …
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 propagat…
While influence maximization in social networks has been studied extensively in computer science community for the last decade the focus has been on the progressive influence models, such as independent cascade (IC) and Linear threshold (LT) models, which cannot capture the reversibility of choices. In this paper, we present the Heat Conduction (HC) model which is a non-progressive influence model with real-world interpretations. We show that HC unifies, generalizes, and extends the existing nonprogressive models, such as the Voter model [1] and non-progressive LT [2]. We then prove that selec…
The influence maximization problem is trying to identify a set of K nodes by which the spread of influence, diseases, or information is maximized. The optimization of influence by finding such a set is an NP-hard problem and a key issue in analyzing complex networks. In this paper, a new greedy and hybrid approach based on a community detection algorithm and a MADM technique (TOPSIS) is proposed to cope with the problem, called, Greedy TOPSIS and Community-Based (GTaCB) algorithm. The paper concisely introduces community detection and the TOPSIS technique, then it presents the pseudo-code of t…
The identification of the minimal set of nodes that maximizes the propagation of information is one of the most relevant problems in network science. In this paper, we introduce a new method to find the set of initial spreaders to maximize the information propagation in complex networks. We evaluate this method in assortative networks and verify that degree-degree correlation plays a fundamental role in the spreading dynamics. Simulation results show that our algorithm is statistically similar, regarding the average size of outbreaks, to the greedy approach in real-world networks. However, our…
Influence maximization is a problem of finding a small set of highly influential users, also known as seeds, in a social network such that the spread of influence under certain propagation models is maximized. In this paper, we consider time-critical influence maximization, in which one wants to maximize influence spread within a given deadline. Since timing is considered in the optimization, we also extend the Independent Cascade (IC) model and the Linear Threshold (LT) model to incorporate the time delay aspect of influence diffusion among individuals in social networks. We show that time-cr…
Social media have a great potential to improve information dissemination in our society, yet, they have been held accountable for a number of undesirable effects, such as polarization and filter bubbles. It is thus important to understand these negative phenomena and develop methods to combat them. In this paper we propose a novel approach to address the problem of breaking filter bubbles in social media. We do so by aiming to maximize the diversity of the information exposed to connected social-media users. We formulate the problem of maximizing the diversity of exposure as a quadratic-knapsa…
Profit maximization (PM) is to select a subset of users as seeds for viral marketing in online social networks, which balances between the cost and the profit from influence spread. We extend PM to that under the general marketing strategy, and form continuous profit maximization (CPM-MS) problem, whose domain is on integer lattices. The objective function of our CPM-MS is dr-submodular, but non-monotone. It is a typical case of unconstrained dr-submodular maximization (UDSM) problem, and take it as a starting point, we study UDSM systematically in this paper, which is very different from thos…
We study the problem of revenue maximization in the marketing model for social networks introduced by (Hartline, Mirrokni, Sundararajan, WWW '08). We restrict our attention to the Uniform Additive Model and mostly focus on Influence-and-Exploit (IE) marketing strategies. We obtain a comprehensive collection of results on the efficiency and the approximability of IE strategies, which also imply a significant improvement on the best known approximation ratios for revenue maximization. Specifically, we show that in the Uniform Additive Model, both computing the optimal marketing strategy and comp…
For the purpose of propagating information and ideas through a social network, a seeding strategy aims to find a small set of seed users that are able to maximize the spread of the influence, which is termed as influence maximization problem. Despite a large number of works have studied this problem, the existing seeding strategies are limited to the static social networks. In fact, due to the high speed data transmission and the large population of participants, the diffusion processes in real-world social networks have many aspects of uncertainness. Unfortunately, as shown in the experiments…
The task of allocating preventative resources to a computer network in order to protect against the spread of viruses is addressed. Virus spreading dynamics are described by a linearized SIS model and protection is framed by an optimization problem which maximizes the rate at which a virus in the network is contained given finite resources. One approach to problems of this type involve greedy heuristics which allocate all resources to the nodes with large centrality measures. We address the worst case performance of such greedy algorithms be constructing networks for which these greedy allocat…
Influence maximization, the fundamental of viral marketing, aims to find top-$K$ 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 individual makes its decision based on the benefit of coordination with its 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 maximizati…
We consider stochastic influence maximization problems arising in social networks. In contrast to existing studies that involve greedy approximation algorithms with a 63% performance guarantee, our work focuses on solving the problem optimally. To this end, we introduce a new class of problems that we refer to as two-stage stochastic submodular optimization models. We propose a delayed constraint generation algorithm to find the optimal solution to this class of problems with a finite number of samples. The influence maximization problems of interest are special cases of this general problem c…
Given a social network with nonuniform selection cost of the users, the problem of \textit{Budgeted Influence Maximization} (BIM in short) asks for selecting a subset of the nodes within an allocated budget for initial activation, such that due to the cascading effect, influence in the network is maximized. In this paper, we study this problem with a variation, where a set of nodes are designated as target nodes, each of them is assigned with a benefit value, that can be earned by influencing them, and our goal is to maximize the earned benefit by initially activating a set of nodes within the…
Influence maximization is a prototypical problem enabling applications in various domains, and it has been extensively studied in the past decade. The classic influence maximization problem explores the strategies for deploying seed users before the start of the diffusion process such that the total influence can be maximized. In its adaptive version, seed nodes are allowed to be launched in an adaptive manner after observing certain diffusion results. In this paper, we provide a systematic study on the adaptive influence maximization problem, focusing on the algorithmic analysis of the scenar…
Influence maximization(IM) problem is to find a seed set in a social network which achieves the maximal influence spread. This problem plays an important role in viral marketing. Numerous models have been proposed to solve this problem. However, none of them considers the attributes of nodes. Paying all attention to the structure of network causes some trouble applying these models to real-word applications. Motivated by this, we present weighted independent cascade (WIC) model, a novel cascade model which extends the applicability of independent cascade(IC) model by attaching attributes to th…
In this paper, we study the Multi-Round Influence Maximization (MRIM) problem, where influence propagates in multiple rounds independently from possibly different seed sets, and the goal is to select seeds for each round to maximize the expected number of nodes that are activated in at least one round. MRIM problem models the viral marketing scenarios in which advertisers conduct multiple rounds of viral marketing to promote one product. We consider two different settings: 1) the non-adaptive MRIM, where the advertiser needs to determine the seed sets for all rounds at the very beginning, and …
This paper examines the problem of adaptive influence maximization in social networks. As adaptive decision making is a time-critical task, a realistic feedback model has been considered, called myopic. In this direction, we propose the myopic adaptive greedy policy that is guaranteed to provide a (1 - 1/e)-approximation of the optimal policy under a variant of the independent cascade diffusion model. This strategy maximizes an alternative utility function that has been proven to be adaptive monotone and adaptive submodular. The proposed utility function considers the cumulative number of acti…
In this paper, we first discuss the definition of modularity (Q) used as a metric for community quality and then we review the modularity maximization approaches which were used for community detection in the last decade. Then, we discuss two opposite yet coexisting problems of modularity optimization: in some cases, it tends to favor small communities over large ones while in others, large communities over small ones (so called the resolution limit problem). Next, we overview several community quality metrics proposed to solve the resolution limit problem and discuss Modularity Density (Qds) …
In this paper, we investigate the discount allocation problem in social networks. 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. What does this mean for a business? Essentially discounts should not just be treated as short term solutions to attract individual customer, instead, allocating discounts to a small fraction of users (called seed users) may trigger a large cascade in a social network. This motivates us to study the influence maximization discount allocation problem: given a soc…
The typical algorithmic problem in viral marketing aims to identify a set of influential users in a social network, who, when convinced to adopt a product, shall influence other users in the network and trigger a large cascade of adoptions. However, the host (the owner of an online social platform) often faces more constraints than a single product, endless user attentions, unlimited budget and unbounded time; in reality, multiple products need to be advertised, each user can tolerate only a small number of recommendations, influencing user has a cost and advertisers have only limited budgets,…
