= 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 probability
$$
p_{k\ell}^{(1)}=\operatorname{logit}^{-1}(\beta_{k\ell}),
$$
so
$$
M_i,M_j\stackrel{\mathrm{iid}}{\sim}
\operatorname{Bin}(n_\ell,p_{k\ell}^{(1)}).
$$
In model 2 the corresponding probability is
$$
p_{k\ell}^{(2)}=\operatorname{logit}^{-1}(\beta_k+\beta_\ell).
$$
Because $k\ne\ell$, the within-college indicator in model 3 vanishes, so $p_{k\ell}^{(3)}=p_{k\ell}^{(2)}$. Consequently models 2 and 3 both give
$$
M_i,M_j\stackrel{\mathrm{iid}}{\sim}
\operatorname{Bin}(n_\ell,p_{k\ell}^{(2)}).
$$
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 \b[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.
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