[thaumasiotes]

> However it turns out that it very predictably happens exactly when the graph gets to a certain size. In the same way as 1000 coin flips very predictably have 450-550 heads.

> What's so cool about the theorem is thst it proves _any_ property you can think of has _some_ sudden threshold like that.

...like what? The example you give, of the number of heads yielded from 1000 coin flips, doesn't have a sudden threshold at any point. What do you mean by a sudden threshold?

[thomasahle]

You are right. We have to define the notion of sudden. It works like this: As n goes to infinity you have 100% chance of 49% heads or more, but 0% chance of 51% or more.

[Cerium]

In the coin flip scenario, the intrinsic properties of a coin result in a threshold of 50% which above a certain scale, is very "sharp" or sudden in the transition. Compare this to a hash table or queuing theory, where a certain amount of capacity works well up to a certain point, then it falls apart. If you can design a random test which will display that property, then you can determine it before the fact without having a way to directly calculate that property, like we do for coin tosses.

[rocqua]

The cases I know tend to be about infinite systems, where you have a critical probability below which the chance of some property is exactly 0. Very often, you get that above that probability the chance is exactly 1 (because of [1]).

In general what I am familiar with is 'percolation theory'.

[1] https://en.wikipedia.org/wiki/Kolmogorov%27s_zero%E2%80%93on...