Probability that a component belongs to a giant component

Question

In a configuration model where each node has a degree 2. What is probability that a node belongs to a component of size n.

Answer 1

I don't know what a "configuration model" is, and this looks more like combinatorial graph theory than statistics. You also need to know whether multiple edges between nodes are allowed, whether loops are allowed, and how many nodes there are.

Let's suppose there are $v$ nodes and that multiple edges and loops are allowed. Start at a particular node $A$ and choose one of its edges. The probability of being a loop (i.e. $A$ belongs to a component of size $1$) is $\frac{1}{v}$. If not, suppose it attaches to some other node; then the probability of the second edge from the second node goes back to $A$ (i.e. $A$ belongs to a component of size $2$) is $\frac{v-1}{v}\times\frac{1}{v-1} = \frac{1}{v}$. And so on.

So the probability $A$ belongs to a component of size $n$ is $\frac{1}{v}$ (for $1 \le n \le v$).