Additive class-effect logistic network model

An additive class-effect logistic network model gives an edge joining classes $k$ and $\ell$ the log odds ${\beta }_k+{\beta }_{\ell }$. A further coefficient multiplying $1_{\{k=\ell \}}$ represents a common within-class log-odds effect.

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 ${\mathbb{R}}^{C+1}$.