Generating sparse connected Erdős–Rényi random graphs Given a random graph $G(n, p)$, where $n$ is the number of nodes and $p$ is the probability of connecting any two edges, it is known that $t = \frac{\ln(n)}{n}$ is a threshold for the connectedness of the graph: if $p$ is greater than $t$ the graph will be almost surely connected, otherwise disconnected. The farther you move $p$ from the threshold the higher the chance that the resulting graph will be disconnected/connected (source: https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model).
I am currently trying to output from an Erdős–Rényi process a graph that is as sparse as possible. In my case $n=10000$ and therefore $t=0.00092$, which means that a connected random graph of 10000 nodes should have on average 46000 edges. Through testing with igraph I could go as low as 37000 edges: $t=0.00074$. Now I'm stuck (because the probability of outputting a sparser graph is too low) and I don't know how to further lower the degree of the graphs. Is there a way to generate sparser random graphs? I know I'm already far below the threshold. In case no solution is found for this model what is the best way to generate a very sparse connected graph?
Brenday McKay suggests an algorithm on mathoverflow. He considers a fixed number of edges, but you can also sample from $G(n,p)$ according to his method. First, calculate $c_{n,m}$ for all $m$. Then sample $m$ in proportion to $p^m (1-p)^{\binom{n}{2}-m} c_{n,m}$. Then use his algorithm.
Gray, Mitchell, and Roughan, Generating connected random graphs, suggest an MCMC algorithm, and analyze it empirically.
Both of these algorithms are for sampling $G(n,p)$ random graphs conditioned on the result being connected. There are also many heuristics for sampling other distributions of random connected graphs in the sparse regime, where usually the focus is on generating some &quot;random-looking&quot; distribution rather than a specific one. For example, you can generate a $G(N,p)$ random graph and pick a connected component, or you can add a random spanning tree to a $G(n,p')$ random graph.