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 fraction) matters and because percolation theory doesn’t capture the idea of time dynamics that are important in understanding epidemics, it’s probably better to start from a more appropriate model.
The statement about percolation is true quite generally, not just for Erdős-Rényi random graphs, but also for the square grid. Above the critical threshold, the giant component is a positive proportion of the graph, and below the critical threshold, all components are finite.
The example I’m thinking about is a non-random graph on the square grid where west/east neighbors are connected and north/south neighbors aren’t. Its density is asymptotically right at the critical threshold and could be pushed over by adding additional west/east non-neighbor edges. The connected components are neither finite nor giant.
If all EW edges exist, you’re really in a 1d situation.
Models at criticality are interesting, but are they relevant to epidemiology? They are relevant to creating a magnet because we can control the temperature and we succeed or fail while passing through the phase transition, so detail may matter. But for epidemiology, we know which direction we want to push the parameter and we just want to push it as hard as possible.
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 fraction) matters and because percolation theory doesn’t capture the idea of time dynamics that are important in understanding epidemics, it’s probably better to start from a more appropriate model.
Maybe look at Limit theorems for a random graph epidemic model (Andersson, 1998)?
The statement about percolation is true quite generally, not just for Erdős-Rényi random graphs, but also for the square grid. Above the critical threshold, the giant component is a positive proportion of the graph, and below the critical threshold, all components are finite.
The example I’m thinking about is a non-random graph on the square grid where west/east neighbors are connected and north/south neighbors aren’t. Its density is asymptotically right at the critical threshold and could be pushed over by adding additional west/east non-neighbor edges. The connected components are neither finite nor giant.
If all EW edges exist, you’re really in a 1d situation.
Models at criticality are interesting, but are they relevant to epidemiology? They are relevant to creating a magnet because we can control the temperature and we succeed or fail while passing through the phase transition, so detail may matter. But for epidemiology, we know which direction we want to push the parameter and we just want to push it as hard as possible.
Not, quite, there are costs associated with pushing the parameter. We want to know at what point we hit diminishing returns.