Last updated on: 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. Table of Contents Why measuring the growth of knowledge matters Different approaches for studying knowledge growth The sociological approach The epidemiological approach The bibliometric and statistical approach Three models of how scientific knowledge grows The cumulative progression model The random selection model Cumulative growth with discontinuities The life cycle of scientific specialties Price and the S-shaped curve Crane’s stages of a research area What happens when a specialty declines 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? How useful was this post? Click on a star to rate it! Average rating 3 / 5. Vote count: 2 No votes so far! Be the first to rate this post. We are sorry that this post was not useful for you! Let us improve this post! Tell us how we can improve this post? https://www.nature.com/articles/s41599-021-00903-w https://www.sciencedirect.com/topics/computer-science/invisible-college https://pmc.ncbi.nlm.nih.gov/articles/PMC4723377/ https://garfield.library.upenn.edu/essays/v4p586y1979-80.pdf https://link.springer.com/chapter/10.1007/978-3-642-23068-4_3 https://arxiv.org/pdf/1911.05363 https://www.qualityresearchinternational.com/socialresearch/invisiblecollege.htm https://www.britannica.com/biography/Thomas-S-Kuhn https://royalsocietypublishing.org/rspa/article/480/2302/20240141/66777/ Debunking -revolutionary-paradigm-shifts-evidence https://link.springer.com/article/10.1007/s11192-011-0429-3 Leave a Reply Cancel reply Informetrics & Scientometrics 1 Information and Measurement Information Revisited Framework for Information Exchange Measurement Techniques Informativeness Standardization of Measurement 2 Measure of Information Information and Entropy Shannon Information Probabilistic Information Properties of Shannon Information Derivation of Shannon Information Formula Normalization Condition Relating Semantic Value to Shannon Type Measures Other Shannon Type Measures of Information Semantic Information Fuzzy Information Measure Other Information Measures 3 Informetrics – Definition, Scope and Evolution Definitions Scope Evolution Summary 4 Sociology of Science and Scientometrics Sociology of Science Growth of Scientific Knowledge Social Organization in Research Areas Approaches of Scientometrics to Sociology of Science Models of Growth of Knowledge 5 Organizations Engaged in Scientometrics and Informetrics Studies Organizations Engaged in or Supporting Scientometrics/Informetrics Studies Websites Research Groups/Discussion Groups Periodical Publications Conferences/Seminars/Workshops/Congresses Individuals Engaged in the Study and Research in Scientometrics/Informetrics 6 Law of Scattering and its Applications Introduction Historical Account Bradford’s Law Verbal Form of Bradford’s Law Applications of Bradford’s Law Graphical Representation of Bradford’s Law Conditions for Bradford’s Law Falling Tail of Bradford Curve: The Groos Droop Ambiguity in Bradford’s Law Fitting Bibliographic Data to Bradford’s Law 7 Rank and Size Frequency Models Representations and Organization of Numerical Data Size – Frequency Approach Rank – Frequency Approach Size – Frequency Models Rank – Frequency Cumulative (Fractional) Models Rank – Frequency Cumulative (Non-Fractional) Models Rank – Frequency Non – Cumulative Models 8 Informetrics Phenomena Terminology and Historical Development Selected Laws of Bibliometrics and Informetrics Informetrics Phenomena in Science Practical Applications of Informetrics 9 Analysis of Library Related Data Necessity for Analytical Studies in Libraries Citation Counting: A Versatile Tool for Journal Selection An Alternative Method of Citation Analysis Selection of New Source Journals to Eliminate Bias Due to Country, and Language Weightage Formula to Correct Citation for Post-War Periodicals Three New Bibliometric Parameters to Re-Rank Scientific Periodicals Garfield’s Methods for Cito-Analytical Studies Librametric Analysis Bibliometric Analysis Informetrics Scientometrics: Its Genesis, Scope, Definition, and Applications 10 User Studies User Studies Questionnaire Method Interview Method Diary Method Observation Method Planning a Survey Classification and Tabulation of Data Analysis of Data Presentation of Results Important User Studies Application of User Studies 11 Laws of Scientific Productivity Scientific Productivity – Influencing Factors Scientific Productivity – Problems in Measurement Scientific Productivity – Distribution Characteristics Lotka’s Law Statistical Distributions or Models Application of Lotka’s Law Goodness-of-Fit Test 12 Growth and Obsolescence of Literature Growth of Literature Obsolescence of Literature Growth Vs Obsolescence of Literature 13 Science Indicators Indicators Towards Science Indicators Historical Aspects Functions of Science Indicators S&T Indicators for the Developing Countries Types of Indicators Validity and Reliability of Indicators Building S&T Indicators Literature Based Indicators Patent Indicators 14 Mapping of Science Cognitive Mapping Journal-to-journal Citation Maps Co-citation Maps Co-word Maps Co-classification Maps Descriptive Mapping 15 Elements of Statistics Data and Its Measurement Graphical Representation Measures of Central Tendency Measure of Variability Correlation and Regression 16 Probability Distributions and their Applications Probability – Definition Random Variables Joint Probability Distribution Conditional Probability Distribution Some Special Distributions Applications of Probability 17 Regression Analysis Simple Linear Regression Multiple Regression Stepwise Regression Regression with Qualitative Explanatory Variables 18 Cluster Analysis and Factor Analysis Introduction Cluster Analysis Factor Analysis Examples of Cluster and Factor Analysis 1 Information and Measurement Information Revisited Framework for Information Exchange Measurement Techniques Informativeness Standardization of Measurement 2 Measure of Information Information and Entropy Shannon Information Probabilistic Information Properties of Shannon Information Derivation of Shannon Information Formula Normalization Condition Relating Semantic Value to Shannon Type Measures Other Shannon Type Measures of Information Semantic Information Fuzzy Information Measure Other Information Measures 3 Informetrics – Definition, Scope and Evolution 4 Sociology of Science and Scientometrics Sociology of Science Growth of Scientific Knowledge Social Organization in Research Areas Approaches of Scientometrics to Sociology of Science Models of Growth of Knowledge 5 Organizations Engaged in Scientometrics and Informetrics Studies Organizations Engaged in or Supporting Scientometrics/Informetrics Studies Research Groups/Discussion Groups Periodical Publications Conferences/Seminars/Workshops/Congresses Individuals Engaged in the Study and Research in Scientometrics/Informetrics

6 Law of Scattering and its Applications Historical Account Verbal Form of Bradford’s Law Applications of Bradford’s Law Graphical Representation of Bradford’s Law Conditions for Bradford’s Law Falling Tail of Bradford Curve: The Groos Droop Ambiguity in Bradford’s Law Fitting Bibliographic Data to Bradford’s Law 7 Rank and Size Frequency Models Representations and Organization of Numerical Data Size – Frequency Approach Rank – Frequency Approach Size – Frequency Models Rank – Frequency Cumulative (Fractional) Models Rank – Frequency Cumulative (Non-Fractional) Models Rank – Frequency Non – Cumulative Models 8 Informetrics Phenomena Terminology and Historical Development Selected Laws of Bibliometrics and Informetrics Informetrics Phenomena in Science Practical Applications of Informetrics 9 Analysis of Library Related Data Necessity for Analytical Studies in Libraries Citation Counting: A Versatile Tool for Journal Selection An Alternative Method of Citation Analysis Selection of New Source Journals to Eliminate Bias Due to Country, and Language Weightage Formula to Correct Citation for Post-War Periodicals Three New Bibliometric Parameters to Re-Rank Scientific Periodicals Garfield’s Methods for Cito-Analytical Studies Librametric Analysis Bibliometric Analysis Scientometrics: Its Genesis, Scope, Definition, and Applications 10 User Studies Questionnaire Method Interview Method Observation Method Planning a Survey Classification and Tabulation of Data Analysis of Data Presentation of Results Important User Studies Application of User Studies 11 Laws of Scientific Productivity Scientific Productivity – Influencing Factors Scientific Productivity – Problems in Measurement Scientific Productivity – Distribution Characteristics Statistical Distributions or Models Application of Lotka’s Law Goodness-of-Fit Test 12 Growth and Obsolescence of Literature Growth of Literature Obsolescence of Literature Growth Vs Obsolescence of Literature 13 Science Indicators Towards Science Indicators Historical Aspects Functions of Science Indicators S&T Indicators for the Developing Countries Types of Indicators Validity and Reliability of Indicators Building S&T Indicators Literature Based Indicators Patent Indicators 14 Mapping of Science Cognitive Mapping Journal-to-journal Citation Maps Co-citation Maps Co-classification Maps Descriptive Mapping 15 Elements of Statistics Data and Its Measurement Graphical Representation Measures of Central Tendency Measure of Variability Correlation and Regression 16 Probability Distributions and their Applications Probability – Definition Random Variables Joint Probability Distribution Conditional Probability Distribution Some Special Distributions Applications of Probability 17 Regression Analysis Simple Linear Regression Multiple Regression Stepwise Regression Regression with Qualitative Explanatory Variables 18 Cluster Analysis and Factor Analysis Cluster Analysis Factor Analysis Examples of Cluster and Factor Analysis Share on Mastodon