22 ms·
Three things: Many, if not most when taken to pedantic conclusions, seemingly-superlinear phenomena are logistic “s-curves”[0]. Even if you’re taking over the
by opportune 3y ago
Three things:
Many, if not most when taken to pedantic conclusions, seemingly-superlinear phenomena are logistic “s-curves”[0]. Even if you’re taking over the entire physical universe, eventually you run out of some limiting growth factor like “total addressable market” or “particles in the the reachable universe.” This isn’t a purely nitpicky point - epidemics and “viral ideas” initially have little interference like this, but when they grow enough they run into interference from immune or previously infected individuals being much harder to infect. Tiktok for example surely had an exponential growth phase among the 14-24 demographic in the US, but by now everyone in that demographic has at least heard of it, and if it did near total penetration into that demographic it’d eventually be limited by the actual number of people able to use their app.
There are also log returns, which look superlinear (not really exponential technically) but eventually taper. For example, a skilled aerospace engineer could probably make a crappy paper plane for $0.1, a really good one for $100. But for $10000 it probably wouldn’t be that much better than the $100 if at all. Plenty of things are like that, it’s called diminishing returns…
Finally, there are definitely quadratic curves in real life. If you’re looking at communications it’s everywhere, usually as an upper bound or worst case. Additional leaf connections may have their diminishing marginal utility countered by the fact that existing and new connections become incrementally more valuable as the network grows.
[0] https://en.m.wikipedia.org/wiki/Logistic_function https://en.m.wikipedia.org/wiki/Logistic_function
- ad93611 3y agoCould you please elaborate on the quadratic curves in real life?
- chrchang523 3y agoSuppose the value of a network to an individual user is proportional to the number of users. Then the total value of the network, summed across all its users, is proportional to the square of the number of users. See also https://en.wikipedia.org/wiki/Network_effect https://en.wikipedia.org/wiki/Network_effect .
- marginalia_nu 3y agoYou see it in processes where something spreads to its vicinity. This isn't really a natural example, but if you draw a square on a sheet of graphing paper. Next iteration you fill in each adjacent square. Repeat this process until you get tired of it. The radius increases linearly at a constant rate, but the area, the number of squares, as a function of each iteration, is growing quadratically. Take a circular forest in a place where there are no fires and no logging. Its rate of growth is proportional to its circumference, which is proportional to its radius. Its area as a function of time is a quadratic function.
- shusaku 3y agoS-curves are basically an approximation for a step function! Interestingly it brings together the two ideas in the article
- jkaptur 3y agoHyper-pedantic note: I think "sigmoid" is the term that describes these curves in the full generality. https://en.wikipedia.org/wiki/Sigmoid_function https://en.wikipedia.org/wiki/Sigmoid_function