Reaction Paper Influence Maximization in Social Networks: A Competitive Perspective

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1 Reaction Paper Influence Maximization in Social Networks: A Competitive Perspective Siddhartha Nambiar October 3 rd, Introduction Social Network Analysis has today fast developed into one of the most important tools used for the study of relations by incorporating network theory. By representing the nodes of the network as the actors (people, countries, goods etc.) and the links between them as the relationships, researchers have been able to develop and study ties between the entities such as friendship, rivalry, organizational position, economic competition etc. Of particular interest in the recent past has been the study of how information or influence contained within a particular node (which can be a person, organization etc.) spreads through a network. The primary goal of this Reaction Paper will be to study influence maximization from a competitive perspective and to identify possible extensions to studies conducted so far so as to mimic real-world scenarios efficiently. The broad idea in this paper is the attempt at maximizing influence from a game theoretic approach such that it is not necessary that one node be dedicated towards one side. This could be put to use to a variety of applications which involve third party brokers or mediators who prefer not to side with either party which is attempting to maximize the influence of its product. Moreover, dynamism needs to be incorporated by modeling the decay of the spread of activation levels over time. Two examples of online bloggers and of viral marketing strategies are mentioned later on in this paper. This paper will be organized into the following sections. In section 2, an outline will be presented of three prominent studies in this regard which incorporate a broad perspective of the aforementioned ideas. In section 3, a critique will be performed of the studies in order to identify the gaps which motivate this paper and other possible extensions. Section 4 will be dedicated towards the brainstorming of possible ideas and to determine the direction that this term project will undertake. 2 Literature Review 2.1 Maximizing the Spread of Influence through a Social Network [1] The paper by Kempe et al. [1] was one of the earliest papers to study the spread of influence through a network. This study was motivated by a fundamental algorithmic problem posed by Domingo and Richardson [2]. A quick summary of this problem is thus : Suppose one were to convince a set of individuals to adopt a new product, and the goal were to attempt to reach as many individuals as possible as a cascaded fall-out from this process, which individuals would one target?

2 The optimization problem in [2] and a number of other studies being NP-hard, this paper focuses on the collection of NP-hard models that have been extensively studied in the social networks community and provides the first provable approximation guarantees for efficient algorithms. Two widely used diffusion models, the Linear Threshold and the Independent Cascade Models are considered. The strategy is essentially the following. An initial set of nodes A o are defined as active to start the diffusion process. The influence of a set of nodes A, denoted by σ(a), is defined as the number of active nodes present at the end. The influence maximization problem then defines a parameter k and attempts to find the k-node set of the largest influence. The primary result of this study is that an optimal solution for influence maximization can be approximated, providing a performance guarantee of about 63%. A natural greedy-hill climbing strategy is the algorithm implemented for the same. The approximation algorithms are also tested against large collaborative networks and it is shown that they outperform other node-selection heuristics involving centrality measures from the field of Social Networks. 2.2 Maximizing Influence in a Competitive Social Network: A follower s perspective [3] Carnes et al. [3] studied the scenario wherein a company wishing to introduce a new product in the market has to compete with a rival company s product. This study is important because it is one of the earliest papers to apply the concept of game theory to influence maximization problems. The paper proposes two models for the spread of influence in the network of consumers. It is assumed that the follower has a fixed budget and it is shown that although selecting the most influential nodes is an NPhard problem, an algorithm which is 63% efficient is possible. An important result of the study shows that it is possible for a follower to select a smaller number of initial nodes and still emerge as the most influential product in the market by the end. From a game theoretic perspective, this problem is similar to determining the company s best response to a competitor s move in a Stackleberg game which is a game where a follower makes his move after a leader does. This study extends the independent cascade model proposed in [1] to the case of two technologies. Consumers are modeled as nodes in a network with the links representing the interactions between them. A set of initial nodes is decided for each of the two technologies, ensuring that there are no overlaps. A third option is also provided which takes care of the scenario in which neither product is adopted. The problem is then solved by two models, a distance-based model and a wave propagation model. In the former, it is assumed that a product may only obtained from an initial adaptor and consumer interested in the technology picks one of the closest available adaptors at random. In the latter model, the technology availability is not limited and the consumers wishing to obtain a product may do so from any of the neighbors who have adopted the product. This study also develops approximation algorithms for the two models apart from running simulations to apply the models on a real-world network of a co-authorship graph based on papers in theoretical high level physics. 2.3 Competitive Influence Maximization in Social Networks [4] Bharathi et al. [4] in their work model innovation diffusion as a game by incorporating multiple competing innovations. This paper attempts at capturing the scenario wherein multiple companies in the market attempt to market their products simultaneously through a network. Apart from providing a mathematically tractable model (Providing a theorem to give a {1 1/e} approximation algorithm for the

3 solution), they also prove that the price of competition is 2 (Providing a theorem to obtain that the expected total number of nodes activated in any Nash Equilibira is at least half the number activated by the best solution with a single player controlling all the initial activations) and discuss first mover strategies which attempts to maximize expected diffusion against perfect competition. While attempting to model the social network, the authors introduce the concept of activation time for each activation attempt. In a directed graph G = (V,E), it is assumed that each edge, e = (u, v) has an activation probability. If node u becomes active at time t, it attempts to activate its currently inactive neighbor v. When this activation attempt succeeds, v also becomes active at time (t + T uv ) where T uv is an exponentially distributed independent continuous random variable. In the influence maximization game, each player selects a set of at most k i active nodes. Thus, with this set remaining active, the influence diffusion begins as described above. Assuming that the sets T 1 T b are the active sets at the corresponding points, the goal of each player is to maximize the expected value of T i. It can be shown by simple examples that this game has no-pure strategy Nash Equilibria. The paper then goes on to study Best response strategies by proving a lemma that if the strategies of all other players are fixed, the payoffs of the player in consideration is a monotone and submodular function of its strategy. First mover strategies are studied in a duopoly with two players called Red and Blue. Given n lines of certain lengths, the red player gets to make k cuts thus creating k+n pieces whose lengths sum up to the original length. The blue player picks the k largest segments and the red player gets the next largest min(n,k) segments. It is then attempted to find the maximum total size of r red pieces over the first i lines. It is shown that the optimal solution cuts each line into equal-sized pieces. This principle can then be extended to a directed line in a directed graph. 3 Paper Discussions The following points in this section will perform a critical evaluation of the above mentioned papers from the point of view of the broad goal described in the introduction. The paper by Kempe et al. [1] is one of the cornerstone studies in the field of Influence Maximization in Social Networks. Much of the literature following this paper derives from or extends on the methods described in this study. However, while the paper describes a general framework for the identification of the key players in the network, it leaves open the question of influence of competing processes or products diffusing through a network. Another assumption that is made in this paper which may not necessarily be valid as a real-world assumption is the one wherein once a new product has influenced a node, it remains so until the end. Very often, one can observe that the effect of the influence does not remain a constant value during an extended period in time. A good example of this is the effect of reading an article about a certain product or idea. Carnes et al. [3] attempts to fill the gap expressed in [1] while creating a model wherein a company wishing to introduce a new model has to compete against a rival company s product. Here, the company attempts to come up with the best response to a competitor s move similar to a stackleberg game. The major significance of this paper is the introduction of a strategic model for the determination of the key players in the network. This is advantageous as it helps in moving towards a real-world scenario because individuals always base their decisions strategically around

4 their rivals, in a very loose sense of the word. Moreover, this attempts to capture the dynamic nature of the network and enables one to model this efficiently. However, the model here is one wherein a strategy is decided by a single player for the network as a whole. This is indicated as there are no overlaps in the product adopted by a node which means that a node can adopt only one product at a time. While this corresponds to a majority of real-life situations, a number of scenarios exist wherein the moderator of the game wishes to study the possible strategic choices of a particular node based on the payoffs made available to it, meaning that it may choose to adopt both products and promote both of them in proportion to what he gains out of both. Moreover, as earlier, the influence remains a constant value throughout the course of the game. The work by Bharathi. et al. [4] is quite important in light of this reaction paper as it incorporates time as a parameter in the maximization model. The significance of time is to capture the spread of influence in a dynamic manner; similar metrics could be incorporated in order to study the decay of influence. The study also extends the number of competing products to a number greater than 2. However, like in [3], the model studies the dynamics of the competition taking place between competing products on a network-wide scale rather than on a node-specific scale. Moreover, the decay of influence has not been considered in this study either. 4 Discussions and Brainstorming The broad field of Influence Maximization is not very old. While a large amount of literature exists on a variety of topics, there is still ample scope to extend existing research. One of the key ideas behind developing models in this topic is to attempt to develop it in as realistic a way as possible. Consider the scenario during election campaigning. Both candidates wish to spread their reach to as many people ( nodes ) as possible. An effective means of doing this is by enlisting the services of online bloggers. Each candidate approaches a blogger (discretely) and requests him to spread a positive effect regarding his candidacy amongst the people who read his blog. The interesting aspect of this game is that the spreading of influence by a blogger is not a binary variable. This is to say that a blogger is not influenced by either candidate. He merely sends out influence signals regarding each candidate in proportion to his payoff from each candidate, (eg., the amount he is paid) Thus, given a fixed budget for each candidate, the game could attempt to find a fixed number of key nodes (nodes of the blogger ) such that the payoffs would be higher for one particular candidate. The important aspect here is that before a move, neither candidate has information regarding which candidate the blogger will publicize more. A possible way of modeling this is by formalizing the game in an extensive form with regards to the actions of each chosen set of bloggers. Another key attribute that hasn t been considered in the past, yet is important, is the decay of activation levels. In the example of election campaigning, consider a node that has been influenced in favor of a particular candidate. With the passage of time, the extent to which this node has been activated is bound to decrease unless the candidate invests more (say, money) in a blogger in order to spread the influence again, thus increasing the level of activation of the node. In the absence of this renewal of activation, an opposing candidate may well be able to shift the support of the node in his favor. Thus, over the course of a certain period until the date of the elections, the candidates would have to ensure that not only have they invested sufficiently in garnering a high amount of support, but they have also managed to ensure that

5 they don t lose their support during the passage of time due to decay in activation levels. Moreover, this decay could also be modeled as a function of distance from the blogger. For instance, if it has to take 5 steps in order for a certain message to reach a certain individual, it can be assumed that the level to which this node is activated is not the same as if it had taken only 2 steps. One of the key issues here would be the inability of obtaining a dataset for the validation of the model. This is owing to the fact that the candidates would never agree to release data of this nature. A different application of a similar model could be in the field of viral marketing. A company might decide to pay a certain amount to an organization to promote its product. However, the level of promotion would be solely based on the amount the company offers to pay. In the meantime, a rival company might decide to also pay a certain amount to the same organization in order to promote its product. In this way, a strategic game develops between the two companies. After studying this model, we would also introduce the concept of decay time and study the change this makes to the model. One possible means of developing this model from a game theoretic approach is by creating a mulltidimensional payoff matrix for all possible combinations of blogger nodes. Thus, if we had n blogger nodes, we would have 2 n possible combinations on the matrix. Then,, given a fixed budget for each candidate, we would run a maximization problem on the overall set of nodes in the case of competing products. This would then give us an idea of the minimum amount of money a candidate would have to pay each of the bloggers in order to ensure they may win the support of a majority of people and win the elections. It can also be studied if a pure strategy Nash Equilibrium exists for this problem. In order to mathematically formulate and solve the model, the formulation employed in [3] could be used. Incorporating the decay of influence in this model would be accomplished in a number of possible ways. Similar to the Linear Threshold model [1], it can be assumed that a node has a certain threshold value for activation. A parameter is defined which refers to the level of activation. Once a node is activated, the value of this parameter associated with this node decreases (possibly exponentially) until it is in a position to be activated again, either by the same candidate or a different candidate. Optimization via Markov Chain Monte Carlo method is another option which can possibly be looked into. The state of the system could be represented by the state of the adjacency matrix at a certain point in time and the rates at which the states change would correspond to the rate at which the bloggers manage to influence people or the rate at which the value of the parameter corresponding to level of activation decreases. 5 Conclusions This reaction paper has provided with a number of possible ideas that could be explored during the course of this project. The gaps identified are consistent with the scope of this paper and it would be quite worthwhile to study them and to fill the gaps. As a next step, the development of a model could be undertaken which captures the necessary attributes including decay of influence and strategic competition in a network. It can also be attempted to formulate an optimization problem using any of the approaches mentioned in [1], [3] or [4] as a starting point and by incorporating possible suitable extensions.

6 6 References [1] D. Kempe, J. Kleinberg, and E. Tardos. Maximizing the spread of influence through a social network. In Proc. of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining. ACM, [2] M. Richardson, P. Domingos. Mining Knowledge-Sharing Sites for Viral Marketing. Eighth Intl. Conf. on Knowledge Discovery and Data Mining, [3] T. Carnes, C. Nagarajan, S. M. Wild, and A. van Zuylen. Maximizing influence in a competitive social network: a follower s perspective. In ICEC 07, pages , New York, NY, USA, ACM. [4] S. Bharathi, D. Kempe, and M. Salek. Competitive influence maximization in social networks. In WINE, pages , 2007