# Expected size of the connected component containing a randomly selected node

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?

• Is p a constant, or a function of n? – Studentmath Sep 30 '14 at 11:07
• p is some constant independent of n. Ideally I'd like to know the expected size of the connected component containing a randomly selected node as a function of p and n. – user83704 Sep 30 '14 at 11:15
• If n is arbitrarily large, the graph is w.h.p connected and the question loses its point. When n is also a constant, it requires some thinking. Will try to. – Studentmath Sep 30 '14 at 11:54
• yeah, I'm interested in the behavior for arbitrary finite values of n. Thanks! – user83704 Sep 30 '14 at 12:06
• Just so you know, by symmetry, the distribution of the component containing a randomly selected node is the same as the distribution of the component containing some generic fixed vertex. – D Poole Oct 3 '14 at 12:49

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:

$\sum\limits_{k=1}^{n} P(k, p, n) \times k$