Bradley-Terry-Luce (BTL) model estimation is a well-established strategy to rank a collection of items given a dataset of pairwise comparisons. Although the theoretical performance of BTL estimation methods, such as spectral and maximum likelihood estimation, is well studied in the regime of uniformly sampled graphs, generalizing such results to a wider class of random graphs has proved challenging. In this work, we investigate the entry-wise error of spectral algorithms against a semi-random adversary that can arbitrarily boost the sampling probabilities of certain edges. We find that the performance of the unweighted spectral method is heavily dependent on the spectral properties of the generated graph. Furthermore, we show that asymptotic performance approaching that of uniformly sampled graphs can be recovered by appropriately reweighting the observed edges to counteract the adversary and restore the spectral gap. Finally, we provide numerical simulations that support our theoretical findings.

翻译：Bradley-Terry-Luce (BTL) 模型估计是基于成对比较数据对项目集合进行排序的经典策略。尽管在均匀采样图场景下，谱估计和最大似然估计等BTL估计方法的理论性能已得到充分研究，但将此类结果推广至更广泛的随机图类别仍具挑战性。本文研究了谱算法在半随机对抗环境下的逐元素误差——该对抗者可任意提升特定边的采样概率。我们发现未加权谱方法的性能强烈依赖于生成图的谱特性。进一步研究表明，通过适当重加权观测边以抵消对抗效应并恢复谱间隙，可渐近达到接近均匀采样图的性能。最后，我们提供了数值模拟以验证理论发现。