Cambridge, MA, March 26, 2026 (GLOBE NEWSWIRE) — When companies seek to grow their capabilities, such as by implementing AI or expanding their services, this might not be easily achieved by simply adding more employees to handle new tasks. New research presents a mathematical model analyzing a wide variety of complex systems — from bacterial cells to corporations to cities — finding, in most cases, the more systems grow, the fewer new functions are added.
The Proceedings of the National Academies of Sciences paper, “Scaling Laws for Function Diversity and Specialization Across Complex Systems,” was co-led by Vicky Chuqiao Yang, assistant professor at the MIT Sloan School of Management and a former Santa Fe Institute (SFI) Omidyar Fellow, and SFI mathematical biologist James Holehouse, along with co-authors Hyejin Youn, José Ignacio Arroyo, Sidney Redner, Geoffrey B. West, and Christopher P. Kempes, all of whom are affiliated with the SFI.
“The perspective of this paper is through a complexity science lens, understanding the interaction between components and how this affects the overall system,” said Yang. “Despite a lot of heterogeneity in the different types of systems we studied, our analysis finds that many systems exhibit common patterns of behavior.”
This work references Heaps’ Law, a formula developed in the field of linguistics, which finds that as the length of text grows longer, the rate at which new words are introduced decreases. The research team analyzed data from bacterial and microbial cells, US federal agencies, companies and universities, and metropolitan areas, and explained the commonalities and differences in the data with a mathematical model for function diversity growth.
Thinking of the distinct functions as the “unique words” and individual systems as “texts,” the researchers found that almost all of these systems followed the same pattern. For example, as a biological cell grows, it tends to produce more of the proteins it already uses rather than developing new ones. Similarly, as a company grows in size, the creation of new job functions slows down and employees will mostly be hired for established jobs rather than new ones.
Yang says this study can shape how companies think about growth and complexity, keeping in mind that increasing the size of systems does not proportionally increase the quantity of different functions within it. As most systems get bigger, the pace at which new functions are added slows at sublinear pace.
“If an organization wants to add a new function category of AI, our research suggests that you can’t just hire a person or people for that role and be done,” said Yang. “To truly expand into a new function, it seems that you need certain infrastructure and existing functions to expand. If you want to become more complex, there’s a foundation you first have to set up.”
Across all of the different types of systems the researchers examined, the function diversity of cities didn’t seem to follow Heaps’ Law, like the others. Function diversity appears to grow more rapidly with population in smaller cities, but slows down as cities become larger. This may be due to the fundamentally different structure and goals of cities.
In addition to looking at “function diversity,” the research identified common patterns in “function abundance.” In corporations and organizations, this represents the number of employees performing each type of job. For example, an organization might have a high concentration of administrative or medical employees. Data showed that these common roles tend to increase more rapidly than less common roles, and remain in abundance over time. The organization grows by hiring more people into these established, abundant jobs rather than creating new ones.
Despite the very different systems analyzed, the researchers’ new mathematical model reveals significant, underlying commonalities of how complexity grows.
“One key takeaway is a striking empirical regularity in the relationship between organizational size and functional complexity,” added Holehouse. “In other words, if an organization aims to be able to do a certain range of tasks, it needs to reach a certain size first.”
- New mathematical model links how cells, organizations, and cities diversify