Because graph theory tells that when the degree is the same and the number of nodes goes lower, it’s more likely to have a network partition (the graph is unconnected).
Network partition is the worst case, but you could also have a connected network where data takes more time to propagate because you need to do more hops given that part of the nodes are down. Take, for example, the figure in your Topic Observation post, and imagine you want to propagate data from node 2 to node 9, but nodes 7 and 8 are down, in those conditions data will need to do three hops instead of two. That is what I meant when I mentioned a decrease in speed if K is too low. If K is larger, you have more mesh connections (e.g., a direct connection between nodes 9 and 1) that still provide some robustness.
Do you have any criteria or formula in mind to determine what K should be?
This is a good question. I think one way to approach this is to start by determining the worst attack we want the network to be able to withstand and then derive the network parameters from there.