Luca Allegri — University of Padua #
Emergent universality as statistical convergence in complex component systems #
Complex component systems are collections of discrete units such as species, words, genes, whose observed realizations are naturally summarized by component counts. Many empirical laws have been observed in those systems, such as Taylor's law, Zipf's law, and Heaps' law. Domain-specific mechanisms are often employed to explain their emergence but, despite their ubiquity, a unifying framework remains elusive. Here we propose such framework showing that, when observables are in the form of component counts, many scaling laws emerge as generic consequences of statistical convergence in large, heterogeneous systems. Taylor's law, for instance, reflects a crossover between sampling noise and genuine system heterogeneity, while Zipf's and Heaps' laws arise from the convergence of order statistics and distinct component counts under heavy-tailed but otherwise generic priors. Moreover, the resulting scaling laws, which we predict theoretically and verify across dozens of complex component systems from biology to linguistics, are largely distribution-agnostic and often transient.
Our work thus suggests that these ubiquitous patterns are better interpreted as baseline statistical limit theorems instead of fundamental principles that require tailored generative explanations.