# Probability that a component belongs to a giant component

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

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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 $1 \le n \le v$).