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Preventing the diffusion of information to vulnerable users while preserving PageRank

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Abstract

Limiting the diffusion of information in social networks is important in viral marketing and computer security. To achieve this, existing works aim to prevent the diffusion of information to as many nodes as possible, by deleting a given number of edges. Thus, they adopt a collective approach and quantify the impact of deletion on the graph, based on the number of deleted edges. In this work, we propose a selective approach which quantifies the impact of edge deletion based on PageRank. Our approach allows specifying the nodes to which information diffusion should be prevented and their maximum allowable activation probability. Furthermore, it performs edge deletion while avoiding drastic changes to the ability of the network to propagate information. To realize our approach, we propose a measure that captures changes, caused by deletion, to the PageRank distribution of the graph. Our measure is called PageRank-Harm (PRH) and quantifies the contribution of an incoming edge \((u_l,u)\) to the PageRank score of the node u. Based on PRH, we define the following optimization problem: Given a subset of nodes and a threshold, find a subset of edges that has minimum PRH and whose deletion limits the activation probability of each specified node to at most the threshold. We show that the problem can be modeled as a Submodular Set Cover (SSC) problem and design an approximation algorithm, based on the well-known approximation algorithm for SSC. Furthermore, we develop an iterative heuristic that has similar effectiveness but also enables significant computational savings. Moreover, we propose a lazy edge selection technique that is used to improve the efficiency of both our approximation algorithm and the iterative heuristic, without affecting their effectiveness. Experiments on real and synthetic data show the effectiveness and efficiency of our methods.

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Notes

  1. The restriction provided in [24] allows u to remain inactive, after each of its active in-neighbors has attempted to activate u.

  2. The value of a monotone measure for the deletion of \(E'\) cannot be larger than that for the deletion of \(E'\cup e\).

  3. This function can be written as \(\sum _{v\in G}(\mathcal {P}(v,Q_{S,v},E))-\sum _{v\in G}(\mathcal {P}(v,Q_{S,v},E'))\), which is supermodular. This is because: (I) \(\sum _{v\in G}\mathcal {P}(v,Q_{S,v},E)\) is constant, (II) \(\sum _{v\in G}(\mathcal {P}(v,Q_{S,v},E'))\) is submodular (as a sum of submodular functions [27]), which implies that \(-\sum _{v\in G}(\mathcal {P}(v,Q_{S,v},E'))\) is supermodular [25], and (III) the sum of a constant and a supermodular function is supermodular [27].

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Authors and Affiliations

  1. Department of Informatics, King’s College London, London, UK

    Grigorios Loukides

  2. School of Computer Science and Informatics, Cardiff University, Cardiff, UK

    Robert Gwadera

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  1. Grigorios Loukides
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  2. Robert Gwadera
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Correspondence to Grigorios Loukides.

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This paper is an extended version of the paper “Limiting the diffusion of information by a selective PageRank-preserving approach” that was presented at the IEEE International Conference on Data Science and Advanced Analytics (DSAA) 2016.

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Loukides, G., Gwadera, R. Preventing the diffusion of information to vulnerable users while preserving PageRank. Int J Data Sci Anal 5, 19–39 (2018). https://doi.org/10.1007/s41060-017-0082-x

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