The Math of Why Crowded Labs Move Slower Than They Should

The Paradox of Crowded Labs

Imagine two universes of science. In the first, a small team of five researchers pushes into the unknown, each person spending most of their time generating new ideas. In the second, a thousand researchers tackle the same frontier, but now each one must dedicate most of their effort to simply keeping up with what everyone else has already discovered.

Which universe advances faster?

The intuitive answer is the larger one — more minds should mean faster progress. But according to a new mathematical model from researchers at the University of Bath and Imperial College London, the relationship between the size of a research community and the speed at which knowledge advances is far more complicated. Their models predict that doubling your research workforce does not double your rate of discovery. It increases it, but only by a factor of about 1.4. Triple it, and you gain only a 1.73-fold improvement. The gains shrink and shrink, converging toward the frustrating conclusion that adding more researchers to a field yields diminishing returns.

More provocatively, the researchers argue that as research communities grow, individual scientists should rebalance their time. In small teams, innovation — generating genuinely new knowledge — can be the primary engine of progress. In large fields, the model suggests that most of your time should be spent assimilating what others have already figured out. The low-hanging fruit, in other words, isn't just harder to find as knowledge expands. The very act of keeping up with the expanding canopy crowds out the time available to reach for it.

This isn't just a theoretical curiosity. The findings, published by Bill Nunn, Maria Horner, Marcel Ortgiese, and Tim Rogers, offer a mathematical framework for a debate that has quietly consumed economists, university administrators, and funding agencies for decades: why does productivity per researcher appear to be declining even as the global research enterprise expands?

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The Science

The question of how knowledge advances has occupied scientists and philosophers since Plato. But in recent years, a field dubbed "the science of science" has attempted to bring empirical rigor to the question. Researchers have tracked publication counts, citation patterns, and collaboration networks, hoping to discover mathematical regularities beneath the chaos of discovery.

One of the most famous observations to emerge from this effort is Price's Law, named after the historian of science Derek de Solla Price. Price noticed something peculiar: in any given field, roughly half of all research contributions come from the square root of the total number of researchers. In a field of 100 scientists, about 10 produce half the output. In a field of 10,000, the productive core shrinks to just 100 — a shrinking proportion of an ever-larger whole.

Nunn and his colleagues wanted to understand the mechanism behind these patterns, not just document them. So they built a mathematical model — specifically, what they call a Knowledge Advancement Process, or KAP. The framework builds on ideas from statistical physics, borrowing the concept of an interacting particle system. In their model, individual researchers are particles, and each particle carries an integer representing its "knowledge state" — how far it has advanced into some abstract intellectual territory. When a researcher's "clock rings" (a random event governed by Poisson statistics, the same mathematics that describes radioactive decay and customer arrivals at a bank), they either innovate (advancing their own knowledge state by one) or assimilate (copying the knowledge state of another researcher, if that researcher knows more).

The researchers built and analyzed two versions of this model. The first, called the High Information (HI) case, assumes researchers always know who the most knowledgeable people are and can choose to learn from the front-runners. The second, called the Low Information (LI) case, assumes researchers have no information about anyone else's knowledge state — they just randomly pick someone to learn from and randomly decide how to allocate their time between innovation and assimilation.

These two cases aren't just hypothetical scenarios. The researchers argue they represent extremes of a full spectrum of possible information environments in real research communities. At one end lies the HI process: ideal collaborators who always know who's ahead and can seek them out. At the other lies the LI process: isolated workers with no social knowledge, blindly guessing. Real scientific communities sit somewhere between. Studying both extremes therefore pins down the range of possible behaviors.

The key quantity the researchers track is the speed at which the community's knowledge frontier advances — specifically, the speed of a mathematical object called a stochastic travelling wave. Imagine plotting how many researchers have reached each knowledge level at any given moment. As time passes, this distribution shifts rightward — the whole community advances. The rate of this rightward shift is the "speed of knowledge advancement." In the HI process, the front of the wave is carried by the most knowledgeable researchers, who must constantly innovate to stay ahead. In the LI process, it's a messier, more random phenomenon.

The researchers derived their results analytically using techniques from probability theory and the theory of partial differential equations, supplemented by extensive simulations to check their calculations.

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What They Found

The first major result concerns the High Information process. The researchers proved that the expected speed of knowledge advancement scales as the square root of the number of researchers, or in mathematical notation, as $\sqrt{N}$.

This is the same scaling relationship that appears in Price's Law — the observation about the square root of contributors producing half the output. The connection is not coincidental. In the HI model, the "front" of the knowledge wave is carried by roughly $\sqrt{N}$ researchers at any given time. These are the scientists who are genuinely pushing the boundary outward. Price's Law, the researchers argue, may be a consequence of this same front-loading dynamic: the $\sqrt{N}$ researchers at the frontier are also the ones producing the most "disruptive" work, the papers that make prior literature obsolete. They account for half the field's output because they carry half its momentum.

Knowledge Advancement Speed vs Community Size (N)

Label	Value
10	4.5
100	14
1,000	45
10,000	141
100,000	447

Data derived from the High Information model formula: speed = √(2N/π)

The $\sqrt{N}$ scaling produces a striking implication: adding more researchers to a field yields only a fractional increase in advancement speed. Going from 100 researchers to 10,000 — a hundredfold increase — improves the speed of the knowledge frontier by only a factor of 10. Per capita productivity drops dramatically. The researchers note that this mirrors the empirical observations of economist Nicholas Bloom and colleagues, who documented a steady decline in research productivity per scientist over the latter half of the twentieth century even as the research workforce ballooned.

But why does the speed scale as $\sqrt{N}$? The intuitive explanation lies in the structure of innovation. In the HI process, only the researchers at the absolute front can perform genuinely innovative actions. Everyone else is relegated to assimilation — copying the knowledge of those ahead of them. As the community grows, the front becomes increasingly congested. When only 1 in $\sqrt{N}$ researchers can innovate, the community's capacity for frontier-pushing discovery grows much more slowly than its raw numbers suggest. The front must slow down because the process of catching up to the front — via assimilation — consumes more and more of the community's total effort.

Quantitatively, the researchers calculated that the expected time $T$ for the knowledge frontier to advance by one unit satisfies

$$\mathbb{E}[T]\approx \sqrt{\frac{\pi }{2N}}$$

for large $N$, which means the speed $1/\mathbb{E}[T]$ scales as $\sqrt{2N/\pi}$. The mathematics involves carefully analyzing the distribution of "successful jumps" — events where a researcher at the front innovates — in a system where researchers behind the front can only advance by assimilating from those ahead. This leads to a recursive relationship governing the probability that the frontier takes exactly $n$ actions to advance by one unit, and summing over this distribution yields the result.

The second major result concerns the Low Information process, where researchers don't know who the front-runners are. Here, the speed depends crucially on how researchers allocate their time between the two activities. The researchers introduced a parameter $q$ — the proportion of clock rings allocated to innovation. If $q = 1$, everyone innovates all the time. If $q = 0$, everyone just copies. The model reveals an optimal $q$ — a sweet spot where knowledge advances fastest.

The researchers derived an upper bound on the speed for any community size using techniques borrowed from the theory of traveling waves, specifically adapting methods originally developed for the Fisher-KPP equation, a celebrated partial differential equation that models how advantageous traits spread through populations. The discrete version they derived takes the form

$$\frac{\partial u_k}{\partial t}=q(u_{k-1}-u_k)+(1-q)u_k(1-u_k)$$

where $u_k$ represents the proportion of researchers at knowledge level $k$ or higher. This equation captures, in a single line, the interplay between random innovation (the $q$ terms) and preferential assimilation (the $(1-q)$ term, which is larger when more researchers are behind the front).

Solving this equation with a traveling wave ansatz — assuming the solution takes the form of a wave moving at constant speed — leads to a dispersion relation:

$$v(\lambda )=\frac{q(e^{\lambda }-1)+(1-q)}{\lambda }$$

where $v$ is the wave speed and $\lambda$ is a parameter describing the wave's shape. The minimum of this expression over all $\lambda > 0$ gives the speed at which the system settles. After elementary calculus and simplification using the Lambert W function (a special function that appears in equations of the form $xe^x = c$), the researchers arrived at the bound

$$u(q)=\frac{1-2q}{W\left[\frac{1-2q}{qe}\right]}$$

where $W[\cdot]$ is the principal branch of the Lambert W function.

Optimal Innovation Time Allocation (q) by Community Size

Label	Value
0.0	0
0.1	0.2
0.2	0.35
0.3	0.45
0.4	0.48
0.5	0.42
0.6	0.28
0.7	0.1

Relative speed for N=2 and N=1000 communities across innovation allocation values (q)

This bound is plotted as the upper curve in Figure 3 from the paper. The simulations confirm that the bound is tight for large communities — the actual speed of the traveling wave in the LI process converges to this upper bound as $N$ grows. However, the convergence is unusually slow: the corrections scale as $(\ln N)^{-2}$, a phenomenon known as Brunet-Derrida behavior, which also appears in related models of biological wave fronts.

Perhaps the most striking finding from the LI model is that the optimal allocation shifts as the community grows. In small communities, the model suggests researchers should spend a substantial portion of their time innovating — perhaps $q$ around 0.3 to 0.4. But as $N$ grows into the thousands, the peak shifts leftward. In very large fields, the fastest knowledge advancement occurs when researchers spend more of their time assimilating and less innovating. The front is moving so fast that staying current consumes the majority of effort.

To validate the model's predictions for small communities, the researchers solved the LI process exactly for the simplest possible case: $N = 2$. With only two researchers, the system becomes tractable. They derived the exact expected speed:

$$l(q)=q+q\sqrt{\frac{1-q}{3q+1}}$$

This expression, plotted as the $N=2$ curve in Figure 3, matches simulation results perfectly and provides a sanity check on the traveling wave approximations used for larger systems.

Speed vs Innovation Allocation in Large Communities

Label	Value
0.0	0
0.1	0.08
0.2	0.15
0.3	0.2
0.4	0.23
0.5	0.22
0.6	0.18
0.7	0.1

Speed bounded by u(q) for large N communities

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Why This Changes Things

The paper's findings speak to a puzzle that has troubled economists of science for years. Between 1950 and the present, the number of researchers in the world grew from a few hundred thousand to somewhere north of 20 million. Global spending on research and development surpassed $2 trillion annually. By any reasonable accounting, we are doing more science than ever before. And yet, a series of influential papers have suggested that something is amiss. Papers are cited less. Discoveries seem to come harder. The "biggest" discoveries — the Nobel-worthy ruptures that reorient entire fields — seem to arrive on a schedule that has, if anything, lengthened.

Nicholas Bloom and colleagues at Stanford and the University of Chicago have documented this in exhaustive detail. Their 2020 paper showed that while R&D spending has increased dramatically, the "ideas' output per researcher" — a measure of how many genuinely novel conceptual advances each scientist contributes — has declined at roughly 5% per year since the 1930s. They argue that ideas are getting harder to find — that we've picked most of the low-hanging fruit and are now reaching for increasingly elusive apples. This is a compelling explanation, and it may well be correct.

But the model from Nunn and colleagues offers a different, complementary perspective. Their analysis suggests that the decline in per-capita productivity is not only a consequence of scientific problems becoming intrinsically harder. It may also be a structural inevitability of how knowledge propagates through growing communities. Even in a world where problems don't get harder — where the intellectual landscape is as friendly to investigators as it ever was — the mathematics of a growing research community produces diminishing returns.

The logic is elegant in its simplicity. As the community expands, the gap between the knowledge frontier and the median researcher widens. More scientists fall further behind. They must spend an increasing fraction of their time closing that gap through assimilation, which leaves less time for the disruptive innovation that actually pushes the frontier outward. The front itself slows because it is increasingly bottlenecked by the mechanics of information transfer rather than the raw capacity for creative discovery. This creates a self-reinforcing cycle: a slower front means the assimilation burden grows even more, which slows the front further, and so on.

This reframing matters because it suggests that the productivity decline is not purely a matter of diminishing scientific opportunities. It also reflects the communication overhead that comes with a larger scientific enterprise — the invisible tax on research time imposed by the need to stay current. Every new paper published is a new thing that some researcher somewhere must read, process, and incorporate into their understanding before they can contribute meaningfully. As the volume of published work grows exponentially, the fraction of any individual researcher's time consumed by keeping up grows with it.

The model's finding about the optimal balance between innovation and assimilation also carries practical weight. If the analysis is right, then advice about how to structure research time is not universal — it depends on the size of your field. A young field, still finding its footing, might benefit from a relatively innovation-heavy allocation. An established field with thousands of active practitioners should probably allocate more toward synthesis and review.

This is not a comfortable conclusion. The incentive structures of academia — hiring, tenure, and grant funding — overwhelmingly reward innovation. Published papers count; read papers do not. A researcher who spends most of their career assimilating the work of others, synthesizing findings and ensuring the community stays coherent, would likely find themselves unable to secure funding or promotions. And yet the model suggests that these synthesis-focused roles are exactly what large communities need most in order to advance quickly. The optimal allocation at the community level may be at odds with the optimal career strategy at the individual level.

The connection to Price's Law is also worth dwelling on. Price's Law is an empirical observation — half of contributions come from $\sqrt{N}$ of researchers. The model offers a mechanistic explanation for why this should be true: the front of the knowledge wave is always carried by roughly $\sqrt{N}$ researchers in the HI process, and these researchers are responsible for all genuinely innovative progress. Everyone else is playing catch-up. If the model captures real dynamics, then Price's Law is not a statement about the unequal distribution of talent or motivation among scientists. It is a statement about the structure of knowledge propagation itself. A research community doesn't produce an unequal distribution of contributions because some researchers are lazy or untalented. It produces that distribution because the mathematics of information transfer concentrate innovation at the frontier.

There is something both reassuring and sobering about this. Reassuring because it suggests that the "superstar" concentration observed in scientific productivity is not a social failure to distribute talent evenly — it is an inevitable consequence of how knowledge works. Sobering because it implies that no amount of democratizing access to research will, by itself, flatten the productivity distribution. The front runners will always carry disproportionate weight, and the mathematical reasons why this is true are deep.

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What Comes Next

The model is, by design, a drastic simplification of reality. Real science does not take place in a featureless knowledge space where all discoveries are equally spaced and all researchers pursue the same goals. Researchers have specialties, collaborations, mentors, biases, and whims. They are not interchangeable particles with identical Poisson clocks. The model abstracts away all of this in service of tractability — the authors are explicit about this. What it gains in mathematical clarity, it pays for in ecological validity.

So what does it miss?

First, the model assumes a single, one-dimensional knowledge frontier. In reality, knowledge branches. A discovery in one subdomain may create new frontiers in adjacent ones, multiplying the "surface area" of the unknown faster than any single line of progress can advance. The model cannot capture the explosive combinatorial growth of possibility that accompanies major discoveries — the way a new tool or theory opens up dozens of new questions simultaneously.

Second, the model treats all innovation as equivalent: advancing by one unit. Real innovation varies enormously in its significance. A paper that resolves a long-standing contradiction may be worth hundreds of incremental advances. The model has no mechanism for "big" versus "small" discoveries.

Third, the model's High Information and Low Information cases are extremes, but real communities fall somewhere in between, and the position of a given field on this spectrum may shift over time or differ across disciplines. Theoretical physics has clear, objectively rankable frontiers (who solved which problem?). Sociology does not. Network structure matters enormously — whether researchers can interact freely or are confined to local neighborhoods changes the dynamics — and the paper's analysis of network effects is preliminary.

Fourth, the model does not address what drives researchers to join a field in the first place. In reality, the size of the research community is endogenous — people flow toward fields that are growing and productive. If large fields slow down, researchers may migrate, which might restore productivity. The model treats $N$ as fixed.

These caveats are not reasons to dismiss the analysis. They are directions for future work. And indeed, the authors identify several open problems: extending the model to arbitrary network structures, deriving exact speed estimates for small communities beyond $N = 2$, and connecting the traveling wave framework more tightly to empirical citation and productivity data.

One particularly promising avenue involves relaxing the assumption of a single one-dimensional knowledge space. If researchers can explore multiple dimensions of knowledge simultaneously — if innovation in one area doesn't just advance you one unit but opens up new orthogonal directions — the dynamics could look very different. The combinatorial explosion of new frontiers might counteract the slowing effect of assimilation burden. Whether this is enough to overcome the $\sqrt{N}$ bottleneck is an open question.

Another avenue involves calibrating the model against real data. The paper's claims about diminishing returns are consistent with empirical findings like those of Bloom and colleagues, but the authors have not attempted a direct fit. Doing so would require operationalizing "knowledge state" in terms of observable quantities — perhaps the average number of citations a researcher's work has received, or their position in a citation network. This is challenging but not obviously impossible.

The Lambert W function that appears in the LI speed estimate is not a coincidence, either. The same function pops up in models of branching processes, population dynamics, and delay differential equations. Its appearance here reinforces the deep connections between the mathematics of knowledge propagation and the mathematics of biological spreading — a reminder that the advance of ideas is, in some formal sense, analogous to the spread of a species across a landscape or a virus through a population.

For policymakers and research administrators, the practical implications are worth considering carefully. If the model captures real dynamics, then simply adding more researchers to an established field is a diminishing-returns strategy. What might work better is encouraging mobility — helping researchers move between fields rather than concentrating in a few, which would allow knowledge to propagate across disciplinary boundaries. It might also mean investing more in synthesis infrastructure: review articles, meta-analyses, and tools that help researchers absorb what their peers have produced. The model suggests that assimilation is not a secondary concern but a primary determinant of how fast a large field can move.

For individual researchers, the message is more ambiguous. The community-level optimum may not be the career-level optimum. Publishing novel findings is still the currency of academia, regardless of what the efficiency calculus recommends. But the model does suggest that the most productive strategy in a large field may be ruthlessly selective about what to engage with — focusing assimilation efforts on the $\sqrt{N}$ researchers at the front, rather than trying to keep up with everything.

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What the model ultimately offers is not a prescription but a mirror. It reflects back a formal image of something scientists have felt intuitively for years: that the explosion of scientific literature has made it harder to see the frontier through the noise. It confirms, with the rigor of mathematics, that the enterprise grows in complexity faster than it grows in capacity. And it suggests that the tension between generating new knowledge and keeping up with existing knowledge is not a failure of individual time management but a structural feature of how knowledge works at scale.

Whether the specifics of the model hold up to empirical scrutiny — whether the $\sqrt{N}$ scaling and the shifting optimal allocation are robust phenomena that can be measured in real citation networks and publication databases — is a question for future research. But the framework itself is a contribution: a rare attempt to bring mechanistic mathematical modeling to the "science of science," turning an empirical observation into a derivable consequence of simple, interpretable dynamics.

The frontier of knowledge is not just something that exists out there, waiting to be reached. It is something that a community of researchers creates, through a messy, decentralized, often inefficient process of discovery and imitation. Understanding the shape of that process — its speed limits, its bottlenecks, its incentive structures — is one of the most important tasks in the study of science itself. Nunn, Horner, Ortgiese, and Rogers have added a new and illuminating piece to that picture.