I am studying a problem that I can model as a random graph. In the basic model, I have a set of vertices that I connect by adding edges. At each stage, I randomly select two vertices and add an edge between them, so an edge could be added more than once. In this case I can either give it a higher weight or just ignore the fact that it was chosen twice - it doesn't matter for my application.
I am interested in the number of connected components as a function of the number of edges I added. What does this distribution look like?
I would like to answer questions like "how many edges do I need to pick to have at least one connected component of size $S$ with probability smaller than $\epsilon$ ?"
I should mention that I am using this for application to a real-life problem, so I am less interested in asymptotic behavior. Specifically, generating edges is associated with a "cost" so I would like to know how many edges I need to generate in order to achieve a given connectivity.
Does anyone have some pointers to relevant material? I found a lot of material on random graph connectivity but would appreciate some material directly relevant to my question, as it is seems pretty straightforward.