Empirical Analysis of Hierarchy and Shared Macroscopic Organization in Soccer Leagues.

This paper presents a coarse-grained framework for analyzing the long-term structural evolution of soccer leagues. This framework draws on ideas from non-equilibrium statistical physics. It combines latent-strength modeling, interaction rules, and large-scale observables to explore the rise of hierarchical organization in competitive leagues. Teams interact based on a Bradley-Terry-type stochastic model. This model relies on teams' latent competitive strengths, which are derived from match outcomes employing maximum-likelihood estimation. Experiments on 38 soccer leagues around the world show persistent hierarchical organization. This is evident in varied latent strengths, uneven competitive balance, and stable macro-state organization from season to season. Even with differences in history, geography, and competition format, many leagues display surprisingly similar large-scale structural patterns. These findings support the idea of shared principles that guide the evolution of competitive leagues and highlight the value of using a statistical physics perspective to describe how macro-level organization arises from repeated team interactions.