The exponential random graph model is defined as, $$ P_\theta(Y=y)=\frac{\exp\{\theta^ts(y)\}}{c(\theta)}.$$ Where $y \in \mathcal{Y}$ the set of all possible networks, $\theta = (\theta_1,\ldots,\theta_s)^t$ is a vector of parameters, $s(y)$ contains the network information (eg. numbers of edges, triangles etc.) and $c(\theta)$ is the normalization factor.
I can't seem to figure out why this distribution belongs to the exponential family of distributions as I need to say something about the density function and not the distribution.
To clarify the density function $p_\theta(y) = p(y|\theta)$ of $P_\theta(Y=y)$ needs to be written as: $$ p_\theta = h(y)\exp \left\{ \eta(\theta)^tT(y) - A(\theta)\right\}. $$ Could anyone help me with figuring out why the density can be written as an exponential family?
The density function is $P_\theta = p_\theta$ as it is a discrete distribution with respect to the counting measure. Then is it fairly easy to write it out as an exponential family.