Election manipulation on social networks: Seeding, edge removal, edge addition

We focus on the election manipulation problem through social influence, where a manipulator exploits a social network to make her most preferred candidate win an election. Influence is due to information in favor of and/or against one or multiple candidates, sent by seeds and spreading through the network according to the independent cascade model. We provide a comprehensive theoretical study of the election control problem, investigating two forms of manipulations: seeding to buy influencers given a social network and removing or adding edges in the social network given the set of the seeds and the information sent. In particular, we study a wide range of cases distinguishing in the number of candidates or the kind of information spread over the network. Our main result shows that the election manipulation problem is not affordable in the worst-case, even when one accepts to get an approximation of the optimal margin of victory, except for the case of seeding when the number of hard-to-manipulate voters is not too large, and the number of uncertain voters is not too small, where we say that a voter that does not vote for the manipulator’s candidate is hard-to-manipulate if there is no way to make her vote for this candidate, and uncertain otherwise. We also provide some results showing the hardness of the problems in special cases. More precisely, in the case of seeding, we show that the manipulation is hard even if the graph is a line and that a large class of algorithms, including most of the approaches recently adopted for social-influence problems (e.g., greedy, degree centrality, PageRank, VoteRank), fails to compute a bounded approximation even on elementary networks, such as undirected graphs with every node having a degree at most two or directed trees. In the case of edge removal or addition, our hardness results also apply to election manipulation when the manipulator has an unlimited budget, being allowed to remove or add an arbitrary number of edges, and to the basic case of social influence maximization/minimization in the restricted case of finite budget. Interestingly, our hardness results for seeding and edge removal/addition still hold in a reoptimization variant, where the manipulator already knows an optimal solution to the problem and computes a new solution once a local modification occurs, e.g., the removal/addition of a single edge.