 Solution

Let $n_{\ell }$ be the number of graduates in college $\ell$. In model 1, every edge from a person in college $k$ to a person in college $\ell$ has probabilitysoIn model 2 the corresponding probability isBecause $k\ne \ell$, the within-college indicator in model 3 vanishes, so $p_{k\ell }^{(3)}=p_{k\ell }^{(2)}$. Consequently models 2 and 3 both give

The distinct edge indicators used in $M_i$ and $M_j$ are independent under all three models, so every model also forces $M_i$ and $M_j$ to be independent. Thus all members of one college have the same cross-college friendship propensity, with no person-level heterogeneity or dependence between their friendship counts. Real friendship networks can have unusually sociable individuals and correlated choices of friends, producing overdispersion and dependence that these models cannot fit.