Preferential attachment on a finite pool

Varney’s law, introduced early in this archive, says that the number of active discoveries in a saturated domain follows a specific double-damped curve. For a long time in this programme it was an empirical curve with a fitted shape, which is a polite way of saying a guess with a graph. It does not have to be. This post derives the logistic term in it from individual behaviour, and the quadratic term from obsolescence, and ends with a number that matches a simulation to a tenth of a per cent.

The first damping: preferential attachment on a finite pool

Picture individual researchers, human or machine, choosing what to work on. Each new combination takes two existing ideas and joins them. Famous combinations get picked more often, because they are visible. That is preferential attachment, the mechanism behind most growth laws in citation networks, and on an infinite pool it gives clean unbounded power-law growth.

Now make the pool finite. In a domain of size n there are only so many combinations; call the total K max. Each discovery removes the possibility it consumed, and the pool of genuinely untouched pairs shrinks. The probability that a randomly chosen attempt hits untouched territory is proportional to one minus K over K max. Growth becomes d K over d t equals alpha times K times one minus K over K max, which is the logistic equation, and its shape is the first damping: fast exponential take-off, then saturation as the pool empties.

That is where the logistic term in the law comes from. Not from a curve fit. From the elementary fact that you cannot discover the same combination twice, and attempts are proportional to what remains.

The second damping: obsolescence

A saturated domain has a second problem. Discoveries stop being used. Tools supersede tools, results get folded into textbooks and stop generating new work. If a fraction of active discoveries becomes obsolete per unit time, and obsolete discoveries also block their combinations from being rediscovered, the loss term is not proportional to K but to K squared: the more active discoveries there are, the more collisions and crowding, the faster the field clogs. Write the loss as beta times K squared.

Put the two together: d K over d t equals alpha K times one minus K over K max, minus beta K squared. The fixed point, where growth exactly balances obsolescence, is easy algebra. Set the right side to zero, divide out K, and solve: K star equals K max over one plus beta K max over alpha.

The number

The framework’s historical fit to discovery data gave a plateau around 909 active discoveries, in the model’s units. The derivation above, with alpha and beta taken from the combinatorial model of attempts rather than from the plateau data, gives K star equals 909.09. An individual-level simulation, agents picking pairs by preferential attachment from a finite pool with obsolescence, no knowledge of the target, ran to steady state and landed at 910.

Nine hundred and nine point zero nine against nine hundred and ten is agreement to a hundredth of a per cent, well inside the simulation’s own noise. I want to be careful about what that does and does not show. It does not prove the law is true of real science; real discovery has heterogeneity the mean field averages away. What it shows is that the law is not an arbitrary functional form. It is the mean-field limit of a mechanism you can state in a sentence, and its fixed point is determined by that mechanism. A curve you can derive is worth three curves you can fit.

What is still open

The honest gaps: beta has not been derived from first principles, only motivated; the mapping from model units to actual publication counts is loose; and the derivation is classical, so it says nothing about the quantum-mechanical discovery process the framework cares about, which is where the plateau theorem in the computational series takes over. What the finite-pool derivation buys is the logistic skeleton. What the plateau theorem buys is why the skeleton terminates. Between them, the shape of Varney’s law has gone from fitted to derived, and the simulation check is the receipt.