This paper (fast generation of large scale social networks with clustering) mentioned in its proposition 1 that "in a regular graph, the probability of an edge existing in the fast Chung Lu model is the same as that in the original Chung Lu model".
The proof is pretty complicated. I can not understand why it uses $2M-\bar D_k$ in calculation of $P(e_{ij}\mid v_j)$.
Isn't it straightforward that $P(\text{a give edge is from }v_j\text{ to }v_i)=$$P(\text{the first selected node is }v_j\text{ and the second selected node is }v_i)$ $=\frac{d_j}{2M}\frac{d_i}{2M}$ (the selection of two nodes is independent) $=\frac{DD}{4M}$, which is the same as the last second sentence in the proof.
Is my argument correct?