5. [20 points] Suppose that a social network is abstracted by an undirected graph G, which consists of n nodes. We assume that the graph G is generated in a random manner as follows: between any two nodes, an edge is present with probability p, and absent with probability (1 - p). All the pair-wise connections are generated in an i.i.d. manner (i.e., independent across any pair of nodes, and with the same connection probability p). You observe a graph G with n nodes, and your goal is to estimate the connection probability p from the graph. (a) Write down the likelihood function of G for a fixed value of edge connection probability p. (b) Compute the Maximum-Likelihood (ML) estimator (PML) for p using the observed graph G. Simplify your expression as much as possible. (c) Find the mean and variance of PML- (d) Prove that limn . P(PML - p| > e) - 0 for any 6 5 0