Scaling laws in complex component systems as consequences of heterogeneous sampling
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, and domain-specific mechanisms are often employed to explain their emergence but, despite their ubiquity, a unifying framework remains elusive. In this work, we propose a null model showing that, under heterogeneous latent rates and finite sampling, several commonly observed scaling relations can arise without invoking domain-specific mechanisms. Taylor's law, for instance, reflects a crossover between sampling noise and genuine system heterogeneity and it is largely insensitive to the detailed latent distribution, while Zipf's and Heaps' laws arise from the convergence of order statistics and distinct component counts under heavy-tailed but otherwise generic priors. Our work thus suggests that these ubiquitous patterns are better interpreted as a transient sign of statistical convergence instead of fundamental principles that require tailored generative explanations.