Models and Approaches to the Growth of Scientific Knowledge

Science does not expand at a steady, predictable pace. It surges, slows, branches, and sometimes fades away entirely. For anyone studying informetrics and scientometrics, a central question is how to measure and explain this movement: how does scientific knowledge actually grow, and can we model it mathematically? Over the past several decades, researchers have built a rich toolkit of approaches and models to answer exactly that. This post walks through the main ways scholars study knowledge growth, the three classic models that explain how ideas evolve, and the life cycle that scientific fields tend to follow from birth to maturity and decline.

Why measuring the growth of knowledge matters 🔗

Before we can manage research funding, plan libraries, or evaluate the output of an institution, we need to understand how knowledge accumulates. The pioneer of this enquiry was the physicist and historian of science Derek J. de Solla Price, who treated science itself as a measurable object. By counting journals, papers, and authors over time, he showed that science grows exponentially and roughly doubles in size every 10 to 15 years. That single observation transformed the study of science from a matter of opinion into a quantitative discipline. Once growth could be measured, it could also be modelled and predicted.

Different approaches for studying knowledge growth 🔗

There is no single method for studying how knowledge expands. Researchers borrow tools from sociology, epidemiology, and statistics, and each lens reveals something different. A useful way to organise them is to ask what is being tracked: people, ideas, or publications.

The sociological approach 🔗

The sociological approach focuses on the scientists themselves and the social structures they build. Its most influential figure is Diana Crane, whose 1972 book Invisible Colleges argued that knowledge does not grow through isolated geniuses but through communities. She built directly on Price’s idea of the “invisible college”, an informal network of around a hundred researchers who communicate across institutional boundaries to monitor a rapidly moving research front. Crane found that productivity is unevenly distributed, with a small percentage of researchers responsible for a large share of innovations, and that these communities themselves have life cycles. The lesson is that the diffusion of a shared paradigm depends on the social channels through which ideas travel.

The epidemiological approach 🔗

A second approach treats the spread of ideas like the spread of a contagious disease. In a landmark study, William Goffman and Vaun Newill proposed in 1964 that the transmission of scientific information could be described using the same mathematics used to model infectious disease outbreaks. In this framing, a population contains people who are “susceptible” to a new idea, those already “infected” who carry and transmit it, and those “removed” who have moved on. Goffman later applied this epidemic model to the history of mast cell research, showing that the field followed a long latency period before reaching epidemic proportions. The contact rate between scientists turns out to be the key variable that accelerates how fast knowledge diffuses.

The bibliometric and statistical approach 🔗

The third approach follows the published literature itself, inferring the movement of ideas from the growth of papers, authors, and citations. This is the heart of scientometrics. When researchers cite earlier work, they create links that form citation networks, giving us visual and statistical evidence of how knowledge builds on itself. Mathematically, these methods split into deterministic, stochastic, and statistical models of idea diffusion. Deterministic models suit large systems where random fluctuations are minor; stochastic and statistical models capture the irregularity of smaller fields and individual careers. India’s own rising share of global research output, tracked through databases like Scopus and Web of Science, is a practical example of bibliometric growth measurement in action.

Three models of how scientific knowledge grows 🔗

Beyond the question of which data to track lies a deeper one: what shape does the growth of ideas take? Three classic models offer competing answers. They are best understood not as rivals but as descriptions that fit different fields at different times.

The cumulative progression model 🔗

The oldest and most intuitive view is the cumulative model. Here, science develops through the steady, gradual addition of verified facts, each one building neatly on the last. As one analysis puts it, scientific development is seen as the gradual growth of knowledge based on a sum of facts accumulated by scholars, in which erroneous theories are slowly dislodged and replaced by better ones. In this picture, progress is linear and irreversible, like bricks being added to a wall. It captures the experience of “normal science” well, but critics argue it ignores the disruptions that periodically reshape entire disciplines.

The random selection model 🔗

A second model introduces an element of chance and competition, echoing evolutionary thinking. New ideas appear as variations, and only some survive selection by the scientific community while others are discarded. This view has empirical support in citation patterns. Price observed that in basic science, each new batch of papers links roughly half of its citations to a small core of recent work, while the remaining citations appear to be random selections from the existing literature. The model captures something the cumulative view misses: not every contribution feeds the mainstream, and survival is partly a matter of which variations the community chooses to build upon.

Cumulative growth with discontinuities 🔗

The third model accepts that knowledge accumulates, but insists this accumulation is periodically interrupted by sharp breaks. This is the world of Thomas Kuhn, whose 1962 book The Structure of Scientific Revolutions argued that long stretches of “normal science” are punctuated by revolutions. Kuhn questioned the traditional view of science as a gradual, cumulative acquisition of knowledge, proposing instead that accumulating anomalies eventually trigger a crisis that can only be resolved by replacing one paradigm with another. The shift from Newtonian mechanics to relativity and quantum physics is the textbook example. Interestingly, recent large-scale studies suggest that true paradigm-shifting revolutions account for only about 1% of major breakthroughs, with most progress driven by steadily improving methods and tools. The debate between cumulative and discontinuous growth remains very much alive.

The life cycle of scientific specialties 🔗

Models of how ideas grow naturally lead to a related question: do whole fields have a predictable life span? Both Price and Crane argued that they do, and their work gave scientometrics one of its most enduring images, the S-shaped curve.

Price and the S-shaped curve 🔗

Price recognised that exponential growth cannot continue forever. A field that doubles every decade would eventually consume all the scientists on Earth. So early explosive growth must give way to a slowdown, producing a logistic or S-shaped curve. A new specialty starts slowly, accelerates into a phase of rapid expansion in publications and researchers, then approaches a ceiling where growth flattens and the field stabilises. Emerging areas such as artificial intelligence and nanotechnology display the steep early portion of this curve, which is why their literature seems to balloon almost overnight.

Crane’s stages of a research area 🔗

Crane refined this picture by mapping social structure onto each phase of the curve, drawing on a Kuhnian framework. In the earliest stage, a small group of pioneers establishes the foundations of a specialty, communicating mostly through personal contact. As a key discovery or paradigm takes hold, the field enters a phase of rapid recruitment and exponential growth, and invisible colleges form to coordinate the expanding network. Growth then slows to a more linear pace as the field consolidates and controversies emerge. Crane’s central finding, that invisible colleges are remarkably similar in structure and growth pattern throughout science, suggests these stages are general rather than unique to any one discipline.

What happens when a specialty declines 🔗

The final stages of a specialty’s life can take several forms. A mature field may fragment into ever more specialised sub-fields, each pursuing narrower questions. It may be absorbed into a larger neighbouring discipline once its distinct approach is no longer needed. It may fall dormant when it hits a theoretical or methodological dead end and publication rates drop. And occasionally a dormant area is revived when new tools or fresh questions breathe life back into it. This is why Crane argued that invisible colleges grow and fade in step with the central research problem that gave them purpose. Understanding these patterns helps research administrators and librarians anticipate where a field is heading rather than simply reacting to where it has been.

What do you think? If most scientific progress turns out to be cumulative rather than revolutionary, should we rethink how we celebrate “breakthroughs” and allocate research funding? And when you look at a fast-growing field today, can you tell whether it is still climbing the steep part of the S-curve or quietly approaching saturation?

References

1. https://www.nature.com/articles/s41599-021-00903-w
2. https://www.sciencedirect.com/topics/computer-science/invisible-college
3. https://pmc.ncbi.nlm.nih.gov/articles/PMC4723377/
4. https://garfield.library.upenn.edu/essays/v4p586y1979-80.pdf
5. https://link.springer.com/chapter/10.1007/978-3-642-23068-4_3
6. https://arxiv.org/pdf/1911.05363
7. https://www.qualityresearchinternational.com/socialresearch/invisiblecollege.htm
8. https://www.britannica.com/biography/Thomas-S-Kuhn
9. https://royalsocietypublishing.org/rspa/article/480/2302/20240141/66777/Debunking-revolutionary-paradigm-shifts-evidence
10. https://link.springer.com/article/10.1007/s11192-011-0429-3