A Tale of HodgeRank and Spectral Method: Target Attack Against Rank Aggregation is the Fixed Point of Adversarial Game

Rank aggregation with pairwise comparisons has shown promising results in elections, sports competitions, recommendations, and information retrieval. However, little attention has been paid to the security issue of such algorithms, in contrast to numerous research work on the computational and statistical characteristics. Driven by huge profit, the potential adversary has strong motivation and incentives to manipulate the ranking list. Meanwhile, the intrinsic vulnerability of the rank aggregation methods is not well studied in the literature. To fully understand the possible risks, we focus on the purposeful adversary who desires to designate the aggregated results by modifying the pairwise data in this paper. From the perspective of the dynamical system, the attack behavior with a target ranking list is a fixed point belonging to the composition of the adversary and the victim. To perform the targeted attack, we formulate the interaction between the adversary and the victim as a game-theoretic framework consisting of two continuous operators while Nash equilibrium is established. Then two procedures against HodgeRank and RankCentrality are constructed to produce the modification of the original data. Furthermore, we prove that the victims will produce the target ranking list once the adversary masters the complete information. It is noteworthy that the proposed methods allow the adversary only to hold incomplete information or imperfect feedback and perform the purposeful attack. The effectiveness of the suggested target attack strategies is demonstrated by a series of toy simulations and several real-world data experiments. These experimental results show that the proposed methods could achieve the attacker's goal in the sense that the leading candidate of the perturbed ranking list is the designated one by the adversary.