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I'm stuck with this particular question. Can someone explain/help me?

Suppose we construct a graph in $WS(n,k,p)$, starting from the n vertices in a ring, where each vertex is connected to its first $\frac k2$ right-hand and left-hand neighbors. What is the probability that none of the edges in this original graph is redirected during the construction of the ultimate graph?

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If your graph is undirected, you have $n*k$ vertices connected by $\frac{n*k}{2}$ edges. Probability of each edge to be rewired (redirected) is $p$, and probability that each edge won't be rewired is $1-p$. Then, you have $\frac{n*k}{2}$ independent events each with the probability of $1-p$. So, probability that none of your edges will be redirected is: $$(\frac{n*k}{2})^{1-p}$$

Please note that there is also a possibility that your new graph would form a ring lattice where vertices are shuffled.

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