Preference-based Centrality and Ranking in General Metric Spaces

Ranking or assessing centrality in multivariate and non-Euclidean data is difficult because there is no canonical order and many depth notions become computationally fragile in high-dimensional or structured settings. We introduce a preference-based notion of centrality defined through population proximity comparisons with respect to a random reference draw, yielding a metric-intrinsic statistical functional that is well-defined on general metric spaces. Because the induced pairwise preferences may be non-transitive, we map them to a coherent one-dimensional score via a Bradley--Terry--Luce cross-entropy projection, viewed as a calibrated aggregation device rather than a correctly specified model. We develop two finite-sample estimators a convex M-estimator and a fast spectral estimator based on a comparison operator, and establish identifiability and consistency under mild conditions. Simulations and real-data examples, including high-dimensional and functional observations, illustrate that the proposed scores provide stable, interpretable rankings aligned with the underlying preference centrality.