  Given pairwise comparisons between multiple items, how to rank them so that
the ranking matches the observations? This problem, known as rank aggregation,
has found many applications in sports, recommendation systems, and other web
applications. As it is generally NP-hard to find a global ranking that
minimizes the mismatch (known as the Kemeny optimization), we focus on the
Erd\"os-R\'enyi outliers (ERO) model for this ranking problem. Here, each
pairwise comparison is a corrupted copy of the true score difference. We
investigate spectral ranking algorithms that are based on unnormalized and
normalized data matrices. The key is to understand their performance in
recovering the underlying scores of each item from the observed data. This
reduces to deriving an entry-wise perturbation error bound between the top
eigenvectors of the unnormalized/normalized data matrix and its population
counterpart. By using the leave-one-out technique, we provide a sharper
$\ell_{\infty}$-norm perturbation bound of the eigenvectors and also derive an
error bound on the maximum displacement for each item, with only $\Omega(n\log
n)$ samples. Our theoretical analysis improves upon the state-of-the-art
results in terms of sample complexity, and our numerical experiments confirm
these theoretical findings.
