Another month, another rationality quotes thread. The rules are:
- Please post all quotes separately, so that they can be upvoted or downvoted separately. (If they are strongly related, reply to your own comments. If strongly ordered, then go ahead and post them together.)
- Do not quote yourself.
- Do not quote from Less Wrong itself, HPMoR, Eliezer Yudkowsky, or Robin Hanson. If you'd like to revive an old quote from one of those sources, please do so here.
- No more than 5 quotes per person per monthly thread, please.
- Provide sufficient information (URL, title, date, page number, etc.) to enable a reader to find the place where you read the quote, or its original source if available. Do not quote with only a name.
The main question of percolation theory, whether there exists a path from a fixed origin to the "edge" of the graph, is equivalently a statement about the size of the largest connected cluster in a random graph. This can be intuitively seen as the statement: 'If there is no path to the edge, then the origin (and any place that you can reach from the origin, traveling along paths) must be surrounded by a non-crossable boundary'. So without such a path your origin lies in an isolated island. By the randomness of the graph this statement applies to any origin, and the speed with which the probability that a path to the edge exists decreases as the size of the graph increases is a measure (not in the technical sense) of the size of the connected component around your origin.
I am under the impression that the statements '(almost) everybody gets infected' and 'the largest connected cluster of diseased people is of the size of the total population' are good substitutes for eachother.
In something like the Erdös-Rényi random graph, I agree that there is an asymptotic equivalence between the existence of a giant component and paths from a randomly selected points being able to reach the "edge".
On something like an n x n grid with edges just to left/right neighbors, the "edge" is reachable from any starting point, but all the connected components occupy just a 1/n fraction of the vertices. As n gets large, this fraction goes to 0.
Since, at least as a reductio, the details of graph structure (and not just its edge fract... (read more)