I have a graph G(E,V) with known degree distribution $p_k$. I wish to obtain the generating function of the degree distribution over $k$ when following a randomly selected edge while traversing the graph. I'll call that distribution $p'_k$.
My reasoning is that, considering a node $u$, a given other node $v$ has $p_k$ of having $k$ edges, each of which has an independent probability of being connected to $u$, so we must have $p'_k \propto kp_k$ which, including normalization, gives a GF $x\frac{G'_0(x)}{G'_0(1)}$, consistent with this paper I'm trying to follow.
However, I want to exclude the node visited prior to $u$. The paper's author simply divides the GF by $x$, so it becomes $$\frac{1}{G'_0(1)}\sum_k k p_k x^{k-1}$$ I don't understand this reasoning because the exponent refers to the degree of $v$ and it is one of the edges from $u$ that becomes excluded when travelling back is disallowed.