Degree-sum sufficient statistic for an additive logistic network model

For independent binary edges with log odds ${\beta }_{z_i}+{\beta }_{z_j}+{\beta }_0{1}_{\{z_i=z_j\}}$, the natural statistics are the sum of the vertex degrees in each class and the total number of within-class edges. They form a minimal sufficient statistic whenever the natural parameter space contains an open subset of ${\mathbb{R}}^{C+1}$.