A mystery novel, a history book, and a fantasy epic may have little in common in plot or style. But count the words inside them and a strange regularity appears: many new words show up early, then fewer and fewer as the author reuses what has already been introduced.
That pattern, known as Heaps’ law, turns out not to belong to books alone. A new study in PNAS finds that the same rule also describes how many complex systems grow, from living cells and corporations to universities and government agencies — and could even be used to predict how they will change in the future.
The study, led by scientists at the Santa Fe Institute and MIT, doesn’t just document this regularity; it introduces a mathematical model that quantifies how different systems diversify and specialize. It finds that, while systems vary in how much they invest in creating entirely new functions, once those functions exist, their subsequent growth follows a remarkably universal rich-get-richer process.
“What’s striking is that these systems weren’t designed to follow the same rules,” says James Holehouse, a Program Postdoctoral Fellow at the Santa Fe Institute, who co-led the study with Vicky Chuqiao Yang, a former Santa Fe Institute Omidyar Fellow now at MIT. “Yet when you look at how they grow, you see the same tradeoff between adding something new and building on what already exists.”
In the study, researchers focus on what they call “distinct functions” — the different kinds of work a system performs. In a cell, that might mean different proteins. In an organization, it could mean different kinds of jobs. As systems grow, they do add new kinds of work, but they do so more and more slowly over time.
Using their model, the team analyzed dozens of bacterial and microbial cells, more than a hundred U.S. federal agencies, thousands of companies and universities, and hundreds of metropolitan areas. Across most of these cases, the same pattern appeared: as systems got bigger, the pace at which they added new functions steadily slowed — a pattern known as sublinear growth.
In practical terms, sublinear growth means that doubling the size of a system does not double the number of functions inside it. Instead, growth increasingly comes from expanding what already exists. A growing organization hires more people into established jobs before creating new titles. A cell produces more of the proteins it already uses instead of evolving entirely new ones.
“It is remarkable that cells, bureaucracies, and companies, despite obvious differences, all grow their function repertoire with a similar pattern,” says Yang, an assistant professor at MIT Sloan and the Institute for Data, Systems, and Society. “This suggests that the regularity discovered in Heaps’ law applies not only to what humans create, like books, but also to human organizations themselves.”
Cities, however, follow a different version of the same trend. They still add new kinds of jobs as they grow, but they do so much more slowly, following a logarithmic pattern rather than the power-law pattern seen in other systems. Even as populations soar, genuinely new job types become increasingly rare.
That difference reflects a deeper structural divide. Cells, firms, and agencies behave like organisms, with clear boundaries and unified goals. Cities, by contrast, resemble ecosystems shaped by the independent choices of individuals rather than centralized control.
Geoffrey West, a co-author and Santa Fe Institute Shannan Distinguished Professor, adds, “There are underlying regularities shaping how complexity builds, even in systems that look completely different on the surface.”
This material is based upon work supported by the U.S. National Science Foundation under Award No. 2526746.
Read the study "Scaling laws for function diversity and specialization across socioeconomic and biological complex systems" in PNAS (February 12, 2025). DOI: 10.1073/pnas.2509729123