I have been trying to calculate a value that has come up for me in a geography field, which I feel may be an elementary solved problem in combinatorics or a graph theory (or on the flip side, known to be very difficult). I have some maths background (undergraduate degree) but hardly anything in discrete maths, so I have no intuition here.
My problem is to do with connected geographic areas that have been randomly assigned to a certain colour. Ideally, I want to know the expected distribution of the sizes of connected regions of the same colour. It occured to me that the geographic regions and their neighbours could be considered as a graph, like this (I've shown the sizes of the connected areas, the distribution of which I am interested in):
I'm not sure how to formulate this as a graph theory problem, but it could be something like:
Given a connected graph of size m, partitioned into k (not likely to be connected) subgraphs by assigning points at random, what is the expected size of the connected sub-subgraphs of each of the k subgraphs.
I do not need an exact answer (though of course that would be very helpful) but some guidance on how to consider this problem and whether or not it is a 'standard' or 'solved' problem in graph theory would be amazing. It seems from this related question that percolation theory might help me here (in the sense that considering the red graph, the blue nodes have been removed). I'm not certain though because I'm not interested in whether there is an open path (in my case the answer will always be no).