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Scalable Methods for Adaptively Seeding a Social Network

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Scalable Methods for Adaptively Seeding a Social Network Scalable Methods for Adaptively Seeding a Social Network Thibaut Horel Yaron Singer Harvard University Harvard University [email protected] [email protected] ABSTRACT 1.0 In recent years, social networking platforms have developed into extraordinary channels for spreading and consuming in- 0.8 formation. Along with the rise of such infrastructure, there 0.6 is continuous progress on techniques for spreading informa- Probability tion effectively through influential users. In many applica- 0.4 tions, one is restricted to select influencers from a set of 0.2 users who engaged with the topic being promoted, and due Core set Friends to the structure of social networks, these users often rank 0.0 0 1000 2000 3000 4000 5000 6000 low in terms of their influence potential. An alternative ap- Degree proach one can consider is an adaptive method which selects users in a manner which targets their influential neighbors. Figure 1: CDF of the degree distribution of users who liked The advantage of such an approach is that it leverages the a post by Kiva on Facebook and that of their friends. friendship paradox in social networks: while users are often of individuals who can serve as early adopters of a new idea, not influential, they often know someone who is. product, or technology in a manner that will trigger a large Despite the various complexities in such optimization prob- cascade in the social network. lems, we show that scalable adaptive seeding is achievable. In many cases where influence maximization methods are In particular, we develop algorithms for linear influence mod- applied one cannot select any user in the network but is els with provable approximation guarantees that can be grace- limited to some subset of users. As an example, consider fully parallelized. To show the effectiveness of our methods an online retailer who wishes to promote a product through we collected data from various verticals social network users word-of-mouth by rewarding influential customers who pur- follow. For each vertical, we collected data on the users who chased the product. The retailer is then limited to select responded to a certain post as well as their neighbors, and influential users from the set of users who purchased the applied our methods on this data. Our experiments show product. In general, we will call the core set the set of users that adaptive seeding is scalable, and importantly, that it an influence maximization campaign can access. When the obtains dramatic improvements over standard approaches goal is to select influential users from the core set, the laws of information dissemination. that govern social networks can lead to poor outcomes. Due to the heavy-tailed degree distribution of social networks, 1. INTRODUCTION high degree nodes are rare, and since influence maximiza- The massive adoption of social networking services in re- tion techniques often depend on the ability to select high cent years creates a unique platform for promoting ideas and degree nodes, a naive application of influence maximization spreading information. Communication through online so- techniques to the core set can become ineffective. cial networks leaves traces of behavioral data which allow observing, predicting and even engineering processes of in- An adaptive approach. An alternative approach to spend- formation diffusion. First posed by Domingos and Richard- ing the entire budget on the core set is an adaptive two-stage son [7, 25] and elegantly formulated and further developed approach. In the first stage, one can spend a fraction of the by Kempe, Kleinberg, and Tardos [14], influence maximiza- budget on the core users so that they invite their friends to tion is the algorithmic challenge of selecting a fixed number participate in the campaign, then in the second stage spend the rest of the budget on the influential friends who hopefully have arrived. The idea behind this approach is to leverage a structural phenomenon in social networks known as the friendship paradox [9]. Intuitively, the friendship paradox Permission to make digital or hard copies of all or part of this work for says that individuals are not likely to have many friends, personal or classroom use is granted without fee provided that copies are but they likely have a friend that does (\your friends have not made or distributed for profit or commercial advantage and that copies more friends than you"). In Figure1 we give an example of bear this notice and the full citation on the first page. To copy otherwise, to such an effect by plotting a CDF of the degree distribution republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. of a core set of users who responded to a post on Facebook Copyright 20XX ACM X-XXXXX-XX-X/XX/XX ...$15.00. and the degree distribution of their friends. Remarkably, there are also formal guarantees of such effects. Recent work Scalability. One of the challenges in adaptive seeding is shows that for any network that has a power law degree dis- scalability. This is largely due to the stochastic nature of the tribution and a small fraction of random edges, there is an problem derived from uncertainty about arrival of neighbors. asymptotic gap between the average degree of small samples The main result in [27] is a constant factor approximation of nodes and that of their neighbors, with constant proba- algorithm for well-studied influence models such as indepen- bility [18]. The implication is that when considering the dent cascade and linear threshold which is, at large, a the- core users (e.g. those who visit the online store) as random oretical triumph. These algorithms rely on various forms samples from a social network, any algorithm which can use of sampling, which lead to a significant blowup in the in- their neighbors as influencers will have dramatic improve- put size. While such techniques provide strong theoretical ment over the direct application of influence maximization. guarantees, for social network data sets which are often ei- ther large or massive, such approaches are inapplicable. The Warmup. Suppose we are given a network, a random set of main technical challenge we address in this work is how to core users X and a budget k, and the goal is to select a sub- design scalable adaptive optimization techniques for influ- set of nodes in X of size t ≤ k which has the most influential ence maximization which do not require sampling. set of neighbors of size k − t. For simplicity, assume for now that the influence of a set is simply its average degree. If Beyond random users. The motivation for the adaptive we take the k=2 highest degree neighbors of X, then surely approach hinges on the friendship paradox, but what if the there is a set S of size at most k=2 in X connected to this core set is not a random sample? The results in [18] hold set, and selecting S would be a two-approximation to this when the core set of users is random but since users who problem. In comparison, the standard approach of influence follow a particular topic are not a random sample of the maximization is to select the k highest degree nodes in X. network, we must somehow evaluate adaptive seeding on Thus, standard influence maximization would have k of the representative data sets. The experimental challenge is to most influential nodes in X while the approximation algo- estimate the prominence of high degree neighbors in settings rithm we propose has k=2 of the most influential nodes from that are typical of viral marketing campaigns. Figure1 is a its set of neighbors. How much better or worse is it to use foreshadowing of the experimental methods we used to show this approach over the standard one? If the network has a that an effect similar to the friendship paradox exists in such power-law degree distribution with a small fraction of ran- cases as well. dom edges, and influence is measured in terms of sum of de- grees of a set, then the results of [18] discussed above imply Main results. Our main results in this paper show that that the two-stage approach which allows seeding neighbors adaptive seeding is a scalable approach which can dramat- can do asymptotically (in the size of the network) better. ically improve upon standard approaches of influence max- Thus, at least intuitively, it looks as if two-stage approaches imization. We present a general method that enables de- may be worth investigating. signing adaptive seeding algorithms in a manner that avoids sampling, and thus makes adaptive seeding scalable to large In this paper, our goal is to study the potential benefits size graphs. We use this approach as a basis for design- of adaptive approaches for influence maximization. We are ing two algorithms, both achieving an approximation ratio largely motivated by the following question. of (1 − 1=e) for the adaptive problem. The first algorithm is implemented through a linear program, which proves to Can adaptive optimization lead to significant improvements be extremely efficient over instances where there is a large in influence maximization? budget. The second approach is a combinatorial algorithm with the same approximation guarantee which can be easily To study this question we use the adaptive seeding model parallelized, has good theoretical guarantees on its running recently formalized in [27]. The main distinctions from the time and does well on instances with smaller budgets. The caricature model in the warmup problem above is that in guarantees of our algorithms hold for linear models of in- adaptive seeding the core set X can be arbitrary (it does not fluence, i.e.
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