Suppose I were to take a random graph on $n$ nodes, with some probability $\pi$ that each edge exists. If I were to chose some other probability $\rho$, and use that to determine which nodes I deleted from the graph. Is there a nice way to determine if the graph is still connected after doing this?
I am trying to look at diseases moving through a network. I'm simplifying things by assuming no reproduction. I assume the disease has a 100% fatality rate, but am selecting certain nodes to self-destruct when the disease hits them. These nodes are the ones that are removed from the graph. The ideas is that, if removing these nodes disconnects the graph, then any disease will be unable to spread through the entire network.