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

Part of the Lecture Notes in Computer Science book series (LNISA,volume 10660)

Abstract

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 \(N^{1-\varepsilon }\), where N is the number of vertices in the graph. 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.

The authors gratefully acknowledge the support of the National Science Foundation under Career Award 1452915 and AifT Award 1535912.

A full version of this paper is available on arXiv: https://arxiv.org/abs/1710.02827.

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Fig. 1.
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Fig. 3.

Notes

  1. 1.

    Previous work on community detection in networks [29] defines a different, but related stochastic hierarchical blockmodel, where the hierarchy is restricted to two levels.

  2. 2.

    The rationale of sampling thresholds after the seeds selection is to capture the scenario that the seed-picker does not have the full information on the agents in a social network, and this setting has been used in many other works [24, 32].

  3. 3.

    The phrase “cascade model” here, as well as in the abstract and Sect. 1, refers to the description how each vertex is influenced by its neighbors, which is completely characterized by F and \({\mathcal D}\) in the general threshold model.

  4. 4.

    Since, as it will be seen later, each node in the hierarchy tree represents a community and its children represent its sub-communities, naturally, the relation between two persons is stronger if they are in a same sub-community in a lower level.

  5. 5.

    We use the letter A to denote the vertices in a VertexCover instance instead of commonly used v, while v is used for the vertices in an InfMax instance. Since VertexCover can be viewed as a special case of SetCover with vertices corresponding to subsets and edges corresponding to elements, the letter A, commonly used for subsets, is used here.

  6. 6.

    For the assumption \(m>n+\mathsf{k} \), notice that allowing the graph \(\mathsf{G} \) to be a multi-graph does not change the nature of VertexCover, we can ensure m to be sufficiently large by just duplicating edges.

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Schoenebeck, G., Tao, B. (2017). Beyond Worst-Case (In)approximability of Nonsubmodular Influence Maximization. In: R. Devanur, N., Lu, P. (eds) Web and Internet Economics. WINE 2017. Lecture Notes in Computer Science(), vol 10660. Springer, Cham. https://doi.org/10.1007/978-3-319-71924-5_26

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