Dynamic Ranking with the BTL Model: A Nearest Neighbor based Rank Centrality Method

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 tournamentthe 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' ∈T} on a regular grid T ⊂ [0, 1], we aim at recovering the latent strengths of the items w^*_t ∈ R^n at any time t ∈ [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' ∈T} is a sequence of Erdös-Renyi graphs, we provide non-asymptotic l_2 and l_∞ error bounds for estimating w^*_t which in particular establishes the consistency of this method in terms of n, and the grid size |T |. We also complement our theoretical analysis with experiments on real and synthetic data