biostudies-literatureYang VCOmidyar NetworkNRF Global Humanities and Social Sciences Convergence Research ProgramNational Science Foundation (NSF)e2509729123https://www.ebi.ac.uk/biostudies/studies/S-EPMC12912993biostudies-literatureUnknown123(7)Function diversity, the range of tasks individuals perform, and specialization, the distribution of function abundances, are fundamental to complex adaptive systems. In the absence of overarching principles, these properties have appeared domain-specific. Here, we introduce an empirical framework and a mathematical model for the diversification and specialization of functions across disparate systems, including bacteria, federal agencies, universities, corporations, and cities. We find that the number of functions grows sublinearly with system size, with exponents from 0.35 to 0.57, consistent with Heaps' law. In contrast, cities exhibit logarithmic scaling. To explain these empirical findings, we generalize the Yule-Simon model by introducing two key parameters: a diversification parameteProceedings of the National Academy of Sciences of the United States of AmericaScaling laws for function diversity and specialization across socioeconomic and biological complex systems.PMC12912993NAEF-2133863Yang VCHolehouse JYoun HRedner SArroyo JIWest GBKempes CPfalseScaling laws for function diversity and specialization across socioeconomic and biological complex systems.Function diversity, the range of tasks individuals perform, and specialization, the distribution of function abundances, are fundamental to complex adaptive systems. In the absence of overarching principles, these properties have appeared domain-specific. Here, we introduce an empirical framework and a mathematical model for the diversification and specialization of functions across disparate systems, including bacteria, federal agencies, universities, corporations, and cities. We find that the number of functions grows sublinearly with system size, with exponents from 0.35 to 0.57, consistent with Heaps' law. In contrast, cities exhibit logarithmic scaling. To explain these empirical findings, we generalize the Yule-Simon model by introducing two key parameters: a diversification paramete2026-01-01T00:00:00Z2026 Feb2026-07-16T06:52:00.293Z2026-07-09T10:42:12.141ZS-EPMC129129934167830210.1073/pnas.2509729123