Plot a field’s journals against their article output and Bradford’s Law promises a tidy, predictable shape. Yet when researchers actually draw these curves, the neat line often breaks down right at the end. The tail bends downward, falling below where the theory says it should sit. This dip has a name: the Groos droop. It is one of the most discussed anomalies in bibliometrics, and understanding it tells us a great deal about how scientific literature is really organised, and about the limits of our tidy mathematical models.
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What is the Groos droop?
To understand the droop, you first need the curve it deforms. Bradford’s Law of scattering describes how articles on a given subject distribute themselves across journals. A small number of journals carry a large share of the relevant articles, while the rest of the literature is spread thinly across a long list of journals that each contribute only one or two papers. Bradford grouped journals into zones of decreasing productivity, where each successive zone contains the same number of articles as the core but requires many more journals to reach that total.
When you test this graphically, you use what is often called the Bradford bibliograph. You plot the cumulative number of articles on the vertical axis against the logarithm of the cumulative number of journals (ranked from most to least productive) on the horizontal axis. The theory predicts a characteristic shape: an initial rising curve for the core journals, followed by a long, straight central line. This straight portion is the signature of Bradford’s Law working as expected.
The problem appears at the far right. Instead of continuing in a straight line, the curve often deviates downward from linearity at the tail end, sagging below the projected straight line. This downward deflection among the least productive journals is the Groos droop. The phenomenon takes its name from O. V. Groos, who reported it in 1967 while examining the Keenan-Atherton data, finding that the bibliograph did not stay straight all the way to its end.
Why the droop matters for the curve
The droop is not a trivial cosmetic flaw. It strikes at the heart of how reliable Bradford’s Law is as a predictive tool. If the curve were perfectly straight beyond the core, a librarian or analyst could extrapolate from a sample of journals to estimate the total size of the literature, or to predict how many more journals would need to be searched to capture additional articles. The droop breaks this assumption. Because the tail falls below the straight line, naive extrapolation from the central portion overestimates how many articles the peripheral journals actually contribute.
This is why the droop has attracted so much theoretical attention. Bradford curves are known to take several shapes in practice. In reality these curves can adopt at least six different forms, including the S-shaped curve that ends in the Groos droop, rather than the single clean shape the simplest models assume. Some curves even show the opposite behaviour, a rising tail, where the final journals contribute more than expected. The existence of these variations means the basic graphical formulation is an idealisation, and the droop is the most common departure from it.
Reasons behind the droop
Since Groos first reported the dip, scholars have offered competing explanations. The debate is genuinely unresolved in places, and that is part of what makes the topic interesting. The proposed causes fall into a few broad camps, and they are not mutually exclusive.
Incomplete bibliographies
The oldest and most intuitive explanation blames the data, not the literature. The argument runs like this: when you compile a bibliography on a subject, it is easy to capture articles from the well-known core journals because they are indexed thoroughly and obvious to search. But the obscure periphery journals, which each carry only one or two relevant papers, are far easier to miss. A search that stops short, or an indexing service with limited coverage, will systematically undercount the contributions of these marginal journals.
If many peripheral articles go uncounted, the cumulative total at the tail end falls short of what a complete count would show. The curve droops simply because the bibliography is incomplete. This explanation is attractive because it preserves Bradford’s Law intact: the droop becomes an artefact of imperfect data collection rather than a real feature of how literature scatters.
However, this tidy story has been tested and found wanting. A computer simulation study set out to check whether incompleteness alone could generate the droop. The study simulated incomplete data through both weighted and unweighted sampling and then rejected the hypothesis that the Groos droop is caused by incomplete data sets. In other words, deliberately removing articles from a complete data set did not reliably reproduce the characteristic dip. Incompleteness may contribute, but it cannot be the whole story.
The nature of less productive journals
A second explanation looks at the periphery journals themselves rather than at gaps in the data. The journals at the tail are, by definition, the least productive contributors to the subject. They are the journals that publish one stray article on the topic and then nothing more. Their relationship to the core subject is loose and incidental.
Under this view, the droop reflects a real ceiling on how much these marginal journals can contribute. The straight-line model implicitly assumes that the supply of low-productivity journals keeps expanding in a regular way. In practice, the population of journals that might carry even a single relevant article is finite. As you exhaust the genuinely relevant periphery, there are simply fewer new journals left to add, so the cumulative curve flattens and droops rather than continuing its projected climb. The droop, on this reading, is a structural consequence of the literature having edges.
Interdisciplinary mixing
One of the most illuminating explanations connects the droop to the interdisciplinary character of a subject. This idea comes from work by Egghe and Rousseau, who investigated what actions could deflect an otherwise pure Bradford curve. Their starting point was data that followed Bradford’s Law perfectly, with no droop at all, and they asked what would introduce one.
Their answer is subtle. They showed that taking the union of several pure Bradfordian bibliographies can itself produce a bibliography with a Groos droop. When a research topic draws on several distinct disciplines, the combined bibliography is effectively a merger of separate Bradford distributions, each with its own core and periphery. Merging them does not produce another clean Bradford curve; it produces one that droops. The practical conclusion is striking: a Groos droop can always be expected in interdisciplinary bibliographies.
This matters because it reframes the droop entirely. Rather than being a contradiction of Bradford’s Law, the droop becomes something the law predicts once you account for how real subjects span multiple fields. Egghe and Rousseau concluded that the droop can be explained through Bradford’s Law itself, so the two do not contradict each other. The anomaly turns out to be consistent with the underlying theory after all.
Mathematical and structural constraints
More recent work has approached the droop from a purely mathematical direction. Some scholars argue that the deviation arises from a basic constraint that the simplest formulas overlook: you cannot have a fractional number of journals or articles. Journals and articles come in whole units. Recent analysis attributes deviations from the theoretical shape to integer constraints on the number of journals and articles, particularly in fields with very productive core journals where idealised formulas break down at the extremes.
This line of work uses extended versions of the Leimkuhler and Egghe formulas, combined with stochastic models of how literature accumulates, to identify when and why the shape changes. The value of this approach is that it treats the droop not as noise to be corrected but as a predictable feature with identifiable causes. There is a broader point here too: studies have concluded that the droop is not merely an artefact of incomplete searches but an integral part of the scattering process, and that many of the cumulative rank-frequency models proposed for Bradford’s Law actually fail to reproduce it. A good model of literature scattering should generate the droop on its own, not need it patched in afterwards.
Why the droop matters in practice
For students of library and information science, the Groos droop is more than a curiosity about the shape of a graph. Bradford’s Law is a working tool for collection development and journal selection. Libraries use it to identify the core journals they must subscribe to in order to capture most of the relevant literature in a field, while spending scarce budgets wisely. The straight central portion of the bibliograph is what makes this estimation possible.
The droop is a warning that this estimation has limits. If you build subscription decisions on a straight-line projection, you will misjudge the contribution of the peripheral journals. Knowing that the curve droops, and understanding why, lets the analyst correct for it rather than be misled by it. The shape of the curve also carries information: a pronounced droop may signal an interdisciplinary subject drawing on several distinct cores, which has consequences for how an information system or a bibliography should be designed.
The droop also teaches a deeper methodological lesson. Bradford’s Law, like Lotka’s Law of author productivity and Zipf’s Law of word frequency, is one of the classic empirical regularities of bibliometrics. These laws are powerful precisely because they describe real patterns, but they are idealisations. The interesting science often lies in the deviations. The Groos droop is a case study in how a small anomaly at the edge of a graph can open up genuine questions about data quality, the structure of disciplines, and the mathematics underlying our models. Decades of debate over a downward bend in a tail show that even well-established laws repay careful scrutiny.
What do you think? If the Groos droop can be explained as a natural consequence of merging interdisciplinary literatures, should we still treat it as an anomaly at all, or simply as another expected shape of the Bradford curve? And when a library relies on Bradford’s Law for journal selection, how much should the droop change the way core and peripheral journals are valued?
References
- https://link.springer.com/article/10.1007/s00381-014-2481-9
- https://pmc.ncbi.nlm.nih.gov/articles/PMC1250321/
- https://www.mdpi.com/2304-6775/12/4/36
- https://link.springer.com/article/10.1007/BF02095349
- https://link.springer.com/article/10.1007/BF02017105
- https://www.academia.edu/1307666/On_the_Theoretical_Foundations_of_Bradfords_Law
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