[Pakaran]

The effect you're seeing on the coin flip is best understood by seeing that each coin flip is independent, and in no way connected to the past. So, the odds are based on a fair 50/50 per flip, which leaves you with a simple 2^n sample space of one of each binary combination for n flips. The math for that works out easily, and can be plotted.

The emergence of random structure in graphs, however, is different. The chance of a specific structure, such as a cycle of some length or a spanning tree of certain dimension, can go from not particularly likely (<20%) to significantly likely (95%+) in just a single additional node. Those transition thresholds at which the percentage changes in an intuitively surprising way are the subject matter of interest here.

[ineptech]

Eh, the coins weren't important, that's just an easy thing for me to visualize. If one graphs that emergence -if one graphs the number of nodes in a graph against the chance of finding some structure in a graph of that size - I had imagined one would see a smooth curve like one half of a binomial distribution curve. It sounds like you're saying the graph would look discontinuous?

[lupire]

It's as continuous a want discrete function can be, but yes it has a region of explosive growth, sort of like a sigmoid.

Think of all the Jenga games. What is the probability of the tower collapsing on turn T (or T/H, for a tower of size H), graphed as a function of T (or T/H)?

[Keyframe]

Coins example would work here if he asked what are the odds of getting n of a/b in a row or how many n-length rows for few different ns in a number of flips.