= Solution
For $k=1,\ldots,C$, define the class degree sum
$$
T_k(y)=\sum_{i:z_i=k}\ \sum_{j:j\ne i}y_{\min(i,j),\max(i,j)},
$$
and define the number of within-college friendships by
$$
T_0(y)=\sum_{i<j}\delta_{z_i z_j}y_{ij}.
$$
Each between-college edge contributes once to the degree sum of each endpoint's college, while each within-college edge contributes twice to its college's degree sum. The model-3 likelihood can therefore be written
$$
L_3(\beta;y)
=\frac{
\exp\!\left\{\beta_0T_0(y)+\sum_{k=1}^C\beta_kT_k(y)\right\}}
{\displaystyle\prod_{i<j}
\left(1+e^{\beta_{z_i}+\beta_{z_j}+\beta_0\delta_{z_i z_j}}\right)}.
$$
The <Fisher-Neyman factorization theorem> shows that $T=(T_0,T_1,\ldots,T_C)$ is sufficient.
For two networks $y,y'$, the likelihood ratio is
$$
\frac{L_3(\beta;y)}{L_3(\beta;y')}
=\exp\!\left\{
\beta_0[T_0(y)-T_0(y')]
+\sum_{k=1}^C\beta_k[T_k(y)-T_k(y')]
\right\}.
$$
Because the parameters range independently over $\mathbb R^{C+1}$, this ratio is constant in $\beta$ exactly when every displayed difference vanishes. The <likelihood-ratio criterion for minimal sufficiency> therefore gives the answer
$$
\boxed{T(y)=\bigl(T_0(y),T_1(y),\ldots,T_C(y)\bigr).}
$$
This is the <degree-sum sufficient statistic for an additive logistic network model>.
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