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...
Another month, another rationality quotes thread. The rules are: