Suppose we have a graph G with n nodes, initially, all n nodes are separated. At each time step, the probability of growing an edge between any two nodes is p (so each time step can grow multiple edges, the event that an edge grows between any two nodes are independent).
I want to know the distribution of the time step T at which graph grows into a connected graph, is there a way to do it?
If the distribution is complex, is there a way to get expected time and variance of the time?
If that's still complicated, can we find a non-trivial upper bound for expected time and variance?
If anyone knows related results or a direction to look into will be a great help. Thank you!