In the realm of complex systems—ranging from microscopic biological entities to sprawling metropolitan regions—understanding how diversity and specialization evolve as these systems grow remains a fundamental yet elusive challenge. A recent groundbreaking study published in the Proceedings of the National Academy of Sciences meticulously dissects these dynamics through a mathematical lens, revealing that as systems scale up in size, the pace at which new functionalities emerge actually diminishes. This counterintuitive finding refines our understanding of growth and complexity, and reshapes how organizations and entities might strategize expansion.
The collaborative research effort, co-led by Vicky Chuqiao Yang of MIT Sloan School of Management and James Holehouse of the Santa Fe Institute, alongside their multidisciplinary team, advances the dialogue on complexity science by teasing out universal patterns across a startlingly diverse array of systems. From bacterial cells to U.S. federal agencies, corporations, universities, and metropolitan areas, the study demonstrates that despite the heterogeneity of these systems, they conform to shared scaling laws governing functional diversity.
At the heart of their theoretical framework lies a concept adapted from linguistics: Heaps’ Law. Traditionally used to describe how the number of unique words in a text grows sublinearly with text length, this principle elegantly maps onto functional diversity within complex systems. Analogously, as a system expands—be it in size, population, or organizational depth—the introduction of entirely novel functions occurs at a decelerating rate.
This insight disrupts the simplistic assumption that growth directly correlates with increased functional novelty. Instead, growing systems tend to deepen existing functionalities rather than proliferate new ones. For example, a maturing biological cell predominantly produces more of its existing proteins instead of synthesizing new varieties, exemplifying a focus on refinement over innovation in function. Corporations display a similar trend, favoring additional staffing in established roles instead of creating new job categories with each incremental growth.
Such revelations hold profound implications for how we conceptualize organizational and systemic scaling. Rather than indiscriminately increasing headcount or capacity with expectations of spontaneous diversification, strategic efforts to broaden capabilities—especially in technology integration like artificial intelligence—must consider the prerequisite expansion of foundational infrastructure. Yang emphasizes that embedding new functional categories requires not only personnel but also the structural basis to support them.
Intriguingly, however, cities emerge as an exception in this broad comparative analysis. Unlike other systems examined, metropolitan areas exhibit a more complex growth trajectory where smaller cities initially add functions more rapidly with population increases, but this growth tapers off as they become larger. This divergence perhaps reflects the unique socio-economic dynamics and institutional frameworks intrinsic to urban centers, where functions can be both more fluid and more constrained by infrastructural and governance variables.
Beyond function diversity, the study quantifies “function abundance” patterns within organizations and complex systems. Function abundance relates to the quantity of entities performing particular roles, such as the number of administrative personnel within a company. The researchers found consistent scaling patterns where dominant job roles expand disproportionately faster than less common roles, reinforcing functional specialization via concentration rather than proliferation.
Mathematically, these scaling behaviors are captured through a novel model that unifies these observations across biological, social, and infrastructural systems. The model effectively elucidates not only the shared sublinear trajectory of functional growth but also the uneven expansion of function abundance, weaving a comprehensive picture of how complexity evolves systemically.
This work significantly advances complexity science by unveiling a striking empirical regularity between organizational size and functional complexity. Holehouse notes that to accomplish a broad spectrum of tasks, an organization must first attain a certain size threshold, underpinning functionality with scale. This principle challenges traditional views on nimble innovation, suggesting size and infrastructure are prerequisites to complex diversification rather than corollaries.
From a practical standpoint, the research advises policy makers, corporate strategists, and urban planners to rethink growth paradigms. Efforts to scale a system in hopes of multifaceted growth must prioritize building internal capabilities and functional ecosystems that enable and sustain new functions rather than expecting new capabilities to emerge organically from increased size alone.
Moreover, the findings encourage a refined analytical perspective when approaching innovation within complex systems. It becomes crucial to identify whether observed expansion is a matter of function diversification or merely abundance scaling within existing roles, as these have differing implications for resource allocation and strategic planning.
The study’s interdisciplinary methodology—leveraging data analysis across biological datasets, federal employment records, business structures, academic institution roles, and urban demographics—exemplifies the power of quantitative modeling in unmasking universality in complexity. Such approaches can be instrumental in decoding other multifaceted systems across disciplines.
As global systems become increasingly intertwined and multifaceted, understanding these universal laws of functional scaling offers a blueprint for managing complexity more effectively. It equips decision-makers with predictive insights to orchestrate growth trajectories aligned with systemic constraints and opportunities for innovation.
Ultimately, this research bridges foundational theoretical principles with real-world applicability, uniting mathematical elegance with empirical robustness. By revealing how complexity unfolds with size—often constrained, patterned, and dependent on infrastructure—it reframes the narrative of growth for a wide spectrum of systems central to science, business, and society.
Subject of Research: Cells, organizations, metropolitan areas, and complex socioeconomic and biological systems.
Article Title: Scaling laws for function diversity and specialization across socioeconomic and biological complex systems
News Publication Date: March 25, 2026
Web References:
https://www.pnas.org/doi/10.1073/pnas.2509729123
https://planetmath.org/heapslaw
References:
Yang, V. C., Holehouse, J., Youn, H., Arroyo, J. I., Redner, S., West, G. B., & Kempes, C. P. (2026). Scaling laws for function diversity and specialization across complex systems. Proceedings of the National Academy of Sciences. https://doi.org/10.1073/pnas.250972912
Image Credits: Jennifer Tapias Derch
Keywords: Complexity Science, Scaling Laws, Function Diversity, Heaps’ Law, Organizational Growth, Biological Systems, Cities, Mathematical Modeling
