Given an Erdős–Rényi random graph with n nodes and edge probability p, what is the expected number of nodes in the connected component containing a randomly selected node?
In other words, if I randomly select one node in the graph, how many nodes should I expect to be in the connected component it is a member of?
Let $C(n,p)$ be the probability of an $n$-node Erdős–Rényi random graph with edge probability p being connected. From this question we know that:
$C(n,p) = 1$ if $n=1$ (one node graphs are trivially connected)
$C(n,p) = 1 - \sum\limits_{i=1}^{n-1} C(i,p) {n-1 \choose i-1} (1-p)^{i(n-i)}$ otherwise.
Let $P(k, p, n)$ be the probability that our node of interest is the member of an exactly $k$ node cluster in an $n$-node Erdős–Rényi random graph with edge probability $p$. $P$ can be calculated from $C$ as:
$P(k, p, n) = C(k,p) {n-1 \choose k-1} (1-p)^{k(n-k)}$
Then from $P$ we can calculate the desired expected value as:
$\sum\limits_{k=1}^{n} P(k, p, n) \times k$