# Component size distribution in small Erdos-Renyi networks

I'm looking at $\mathcal{G}(n,p)$ (I'll call these Erdos-Renyi networks) where $n$ is, say, at most 10.

I would like to know the probability a random node is in a component of size $m$.

It's sufficient for me to know what the size distribution is of components in small Erdos-Renyi networks. Is there a straight-forward calculation for this?

The best I can think of at the moment is to try to take all connected graphs of size $m\leq n$, and then determine their probability of being a component of an Erdos-Renyi network. It's not obvious to me how to deal with this since I don't know a straightforward way to enumerate all connected graphs of size $m$ (though given such a connected graph, it's straightforward to calculate it's probability of being a component of an Erdos-Renyi network)

• Note that the probability that some given connected graph $H$ is a component of $G(n,p)$ depends only on the number of vertices and edges. So it is the counts that matter, not the graphs themselves, and that is a much easier task. Commented Sep 13, 2018 at 11:35
• I fear most results on component size in ER graphs are for large $n$. But if $n$ is so small, then can't we enumerate all possible graphs with sampling probability, and their component size? Commented Jan 16, 2021 at 20:06