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I have a graph $G$ which has $n$ nodes and $\alpha*(n^2-n)/2$ edges (so the chance of having edge $(i,j)$ is $\alpha$). The graph is connected which means that if we calculate the number of components it results in 1.

I want to randomly select some nodes of the graph and make a new vertex-induced graph (those nodes plus the edges between them from the original graph) so the new sub-sampled graph stay connected with a high probability. I am looking for the largest possible sub-sampling rate for this task (or the smallest graph size which meets the connectivity condition on average). In other words, I am wondering how small could it be while still connected.

I know that under some circumstances there would be a cut vertex or some major vertices which removing them will remove the connectivity in the new graph, but I am looking for the average as I am want to do

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If you start with a binomial random graph with $n$ vertices and edge probability $\alpha$, and then sample $n'$ vertices uniformly at random, then the result is equivalent to just taking a binomial random graph with $n'$ vertices and edge probability $\alpha$.

When $n'$ is large (equivalently, when $\alpha$ is small), the threshold for connectivity is $\alpha \approx \frac{\ln n'}{n'}$ (which can be made more precise), or equivalently $n' \approx \frac1{\alpha} \ln \frac1{\alpha}$.

The subgraph with this many vertices is connected with high probability as $\alpha \to 0$, which might not be exactly the statement you want. If $\alpha$ is constant and $n \to \infty$, then we can take $n'$ to be any function that grows with $n$ and have the subsampled graph be connected with high probability as $n \to \infty$.

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  • $\begingroup$ I am confused! $\alpha$ is fixed and the size of the graph $n$ is fixed as well. I have a random yet connected graph of let's say 500 nodes that has let's say 0.5 percent of the edges possible (or the chance of two nodes being connected is 0.5). I want to see how small (on average) the connected subsampled graph could be. Can I make a vertex-induced subgraph of let's say 40 vertices and be confident that this subgraph is still connected? $\endgroup$ – ameerosein Feb 21 at 23:40
  • $\begingroup$ Now I am also confused. What do you mean, then, by "with high probability"? Usually, in the study of random graphs, it means "with probability tending to 1 as (some parameter) goes to $\infty$". Still, in your case, $\alpha = \frac1{200}$ is much smaller than $\frac{\ln 40}{40}$, so we don't expect a typical $40$-vertex induced subgraph to be connected. $\endgroup$ – Misha Lavrov Feb 21 at 23:56
  • $\begingroup$ Oh sorry I mean $\alpha=0.5$ or 50 percent $\endgroup$ – ameerosein Feb 22 at 20:25
  • $\begingroup$ Then $\frac1\alpha \ln \frac1\alpha = 2\ln2$ is very small, so we don't get precise lower bounds on $n'$ from the asymptotic bounds. However, a random subgraph on just $3$ vertices has a $\frac12$ chance of being connected; a random subgraph on $4$ vertices has a $\frac{19}{32}$ chance, and so on. For $40$ vertices, it is almost certain, because $\alpha = \frac12$ is much larger than $\frac{\ln 40}{40}$. As before, this does not depend on $n$, only on $n'$ (assuming we start with a randomly chosen graph). $\endgroup$ – Misha Lavrov Feb 22 at 20:43

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