[Math] Expected number of connected components in a random graph

Question

For a random graph G(n,p) what is the expected number of connected components? What is the probability distribution of this value?
I'm specially interested in what happens for small values of p, before the giant component emerges.

Answer 1

Have a look at Chapter 11 of Alon and Spencer's The Probabilistic Method. They focus entirely on the case $p = \Theta(1/n)$, and the smallest p they consider is of the form $c/n$, which perhaps counts as "small p" as in your question. In 11.6, they study the smallest case $c < 1$ in detail (they refer to this case as Very Subcritical). The results are always asymptotic as $n\to \infty$. They prove that the size of a component is modeled well by a Poisson Branching Process, so 11.4.2 gives you the probability distribution of sizes of components. Now it's a simple exercise to find the expected number of components: do a branching process to fill up the first component, then do one for the second, etc. until you run out of vertices. Write down how many components you ended up with.