We refers to Statistical mechanics of networks by Park and Newman in the section. Random graphs Considering exponential random graphs with fixed number of vertices n we know only the expected number of edges $\langle m \rangle$. We have the adjacency matrix: $$\sigma_{ij}=\begin{cases} 1 & \text{if i connected to j} \\ 0 & \text{otherwise} \end{cases}$$
To write the degree distribution we can introduce the generating function: $$g(\alpha)= \sum_{k}p_ke^{\alpha k}$$ After some passages: $$g(\alpha)= \frac{\sum_{G \in \mathcal{G}}\prod_{k<l}exp[-(\theta+\alpha)\sigma_{kl}]}{(1+ e^{-\theta})^{\binom{n}{2}}} \stackrel{?}{=} \left(\frac{1+ e^{-(\theta+\alpha)}}{1+ e^{-\theta}}\right)^{n-1}$$ Does someone know how the last equality is possible? In my opinion should be: $$\frac{\sum_{G \in \mathcal{G}}\prod_{k<l}exp[-(\theta+\alpha)\sigma_{kl}]}{(1+ e^{-\theta})^{\binom{n}{2}}} \stackrel{?}{=} \left(\frac{1+ e^{-(\theta+\alpha)}}{1+ e^{-\theta}}\right)^{\binom{n}{2}}$$