For a degree sequence $(d_1,\ldots, d_n)$ with $\min d_i \geq k$ pick a graph with that degree sequence uniformly at random. What's the probability that the graph is k-connected?
I know that for Erdos-Renyi graphs the k-connectivity threshold coincides with with the appearance of degree k vertices, but that doesn't say much about the probability when we only know that the minimum degree is $k$.
This question is related to this question. I'd like to estimate how many graphs you need to sample before picking a biconnected graph.