The argument for O(n) is well formed here.
O(n^2) is not, the core argument is that these grow faster because of compounding. Compounding is fundamentally an exponential process, far larger asymptotically than a quadratic.
The argument for O(n) is well formed here.
O(n^2) is not, the core argument is that these grow faster because of compounding. Compounding is fundamentally an exponential process, far larger asymptotically than a quadratic.
Businesses generally grow following a few patterns. They generally have some TAM to saturate and saturate the TAM at some rate. This rate can be vaguely linear, what I call O(n), or vaguely superlinear, what I call O(n²). The reason I borrow the asymptotic notation is because it implies the growth rate is an upper bound (best case scenario) and generalizes away specific constant factors and sums. The analogy breaks down when you force n or n² imply something numerically specific about your growth rate, or introduce functions with different growth rates like logs or exponentials. For now we will (somewhat unprincipledly) stick with two sole classes.
Article would be better using something like O(linear) and O(≫linear). The big O notation is a useful and memorable metaphor, but the n squared is really confusing. The article also doesn't use Unicode for the notation - which fucks usability (e.g. I used screenshot OCR and reedited).If you have an exponential (revenues) and another exponential of smaller rate (expenses; assume you have profit) then the difference is still an exponential.
Plot the graph of 2^x - 1.99^x to see what I mean. It stays close to the axis for a while and then suddenly shoots up. Follows the trajectory of quite a few companies now that I think of it.
> [2] Perhaps choosing a better two functions could more closely explain the growth dynamics of network effects, which could be more exponential. I think the analogy diminishes in value if you try to directly numerically match it to some growth metric.