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