The given graph represents users' friend relationship in SNS. By assuming the link probability is a constant, then how can I perform maximum likelihood estimation on the link probability?
In different case, if I cluster '2' and '4' to group A, and '1','3','5' to group B, then the probability of a link between same group members is different from the probability of link between different group members. How can I perform MLE on this probability?
If $p$ is the link probability, then there are ${5\choose 2}$ trials with probability $p$ each. The MLE of $p$ in this case is given by the proportion of successes, e.g. $\hat p=\frac{x}{{5\choose 2}}=\frac{5}{10}=\frac 12$, with $x$ the number of observed edges.
If $p$ is the within-group link probability and $q$ is the out-of-group link probability, again we have $4$ trials with probability $p$ success each, and $2\times3=6$ trials with probability $q$ success each. The MLE's are $\hat p=\frac x 4=\frac 4 4=1$ and $\hat q=\frac 1 6$.