Many applications such as recommendation systems or sports tournaments involve pairwise comparisons within a collection of $n$ items, the goal being to aggregate the binary outcomes of the comparisons in order to recover the latent strength and/or global ranking of the items. In recent years, this problem has received significant interest from a theoretical perspective with a number of methods being proposed, along with associated statistical guarantees under the assumption of a suitable generative model. While these results typically collect the pairwise comparisons as one comparison graph $G$, however in many applications - such as the outcomes of soccer matches during a tournament - the nature of pairwise outcomes can evolve with time. Theoretical results for such a dynamic setting are relatively limited compared to the aforementioned static setting. We study in this paper an extension of the classic BTL (Bradley-Terry-Luce) model for the static setting to our dynamic setup under the assumption that the probabilities of the pairwise outcomes evolve smoothly over the time domain $[0,1]$. Given a sequence of comparison graphs $(G_{t'})_{t' \in \mathcal{T}}$ on a regular grid $\mathcal{T} \subset [0,1]$, we aim at recovering the latent strengths of the items $w_t^* \in \mathbb{R}^n$ at any time $t \in [0,1]$. To this end, we adapt the Rank Centrality method - a popular spectral approach for ranking in the static case - by locally averaging the available data on a suitable neighborhood of $t$. When $(G_{t'})_{t' \in \mathcal{T}}$ is a sequence of Erd\"os-Renyi graphs, we provide non-asymptotic $\ell_2$ and $\ell_{\infty}$ error bounds for estimating $w_t^*$ which in particular establishes the consistency of this method in terms of $n$, and the grid size $\lvert\mathcal{T}\rvert$. We also complement our theoretical analysis with experiments on real and synthetic data.

翻译：许多应用（如推荐系统或体育锦标赛）涉及对$n$个物品集合进行成对比较，其目标是通过聚合比较的二元结果来恢复物品的潜在强度或全局排序。近年来，这一问题从理论角度引起了广泛关注，研究者提出了多种方法，并在特定生成模型假设下给出了相应的统计保证。尽管这些方法通常将所有成对比较结果汇总为一个比较图$G$，但在许多实际应用（如锦标赛中足球比赛的结果）中，成对比较的性质会随时间演变。与上述静态场景相比，针对动态场景的理论结果相对有限。本文研究了经典BTL（Bradley-Terry-Luce）模型从静态场景到动态场景的扩展，其假设成对比较结果的概率在时间域$[0,1]$上光滑演化。给定正则网格$\mathcal{T} \subset [0,1]$上的一序列比较图$(G_{t'})_{t' \in \mathcal{T}}$，我们旨在恢复任意时间$t \in [0,1]$上物品的潜在强度$w_t^* \in \mathbb{R}^n$。为此，我们通过局部平均$t$的适当邻域内的可用数据，改进了秩中心性方法（一种静态排序中流行的谱方法）。当$(G_{t'})_{t' \in \mathcal{T}}$为厄尔多斯-雷尼图序列时，我们给出了估计$w_t^*$的非渐近$\ell_2$和$\ell_{\infty}$误差界，从而建立了该方法在$n$和网格大小$\lvert\mathcal{T}\rvert$意义上的一致性。我们还通过真实及合成数据上的实验补充了理论分析。