= Solution
Conditional on the observed colleges, the <stochastic block model> assumes that the friendship indicators are <independent random variables>. For arbitrary linear predictors $\theta_{ij}$, their <Bernoulli distribution> likelihood is
$$
L(\theta;y)
=\prod_{1\leq i<j\leq m}
\left(\frac{e^{\theta_{ij}}}{1+e^{\theta_{ij}}}\right)^{y_{ij}}
\left(\frac1{1+e^{\theta_{ij}}}\right)^{1-y_{ij}}
=\prod_{i<j}\frac{e^{y_{ij}\theta_{ij}}}{1+e^{\theta_{ij}}}.
$$
Thus the three requested likelihoods are
$$
\boxed{
\begin{aligned}
L_1(\beta;y)
&=\prod_{i<j}\frac{e^{y_{ij}\beta_{z_i z_j}}}{1+e^{\beta_{z_i z_j}}},\\
L_2(\beta;y)
&=\prod_{i<j}\frac{e^{y_{ij}(\beta_{z_i}+\beta_{z_j})}}{1+e^{\beta_{z_i}+\beta_{z_j}}},\\
L_3(\beta;y)
&=\prod_{i<j}
\frac{e^{y_{ij}(\beta_{z_i}+\beta_{z_j}+\beta_0\delta_{z_i z_j})}}
{1+e^{\beta_{z_i}+\beta_{z_j}+\beta_0\delta_{z_i z_j}}}.
\end{aligned}}
$$
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