# Expected size of giant component in ER graph

Question in exam paper. Given an undirected G(n, p) graph with p=1.2/n , what is the expected size of its giant component?

Note: I know that there is a giant component because np = 1.2 > 1 and that the other small components are of size O(log n). I think however that this questions wants a real number like 1.2n or something. What is that number and how do you find it?

• In the limit $n\to\infty$, the size of the giant component behaves like $sn$ where $s>0$ solves $$1-s=e^{-1.2s}$$ – Did Apr 12 '18 at 14:05

The basic idea is that the probability that a given vertex ends up in the giant component is equal to the probability that its associated branching process (with mean $np = 1.2$) becomes infinite.
If $q$ is the probability that the branching process goes extinct, then by conditioning on the number of children of the first node, we have $$q = \sum_{i \ge 0} q^i \cdot \left(e^{-1.2} \frac{1.2^i}{i!}\right)$$ which simplifies to $q = e^{-1.2(1-q)}$. Therefore if $r$ is the probability of being in the giant component, then it is the nonzero solution to $1-r = e^{-1.2r}$, and $nr$ vertices of the graph end up in the giant component.