The directed graph in Fig. 2 depicts the hyperlinks among six Web pages A-E, and also a proposed PageRank value for each page. Properties of PageRank equilibrium values. Are these correct (limiting) equilibrium values for the basic PageRank update rule? Give a brief (one- to three-sentence) explanation for your answer. PageRank and Markov chains. We have seen that limiting PageRank values correspond to the stationary distribution pi of a Markov chain (MC), which describes the sequence of page visits of a random Web surfer. Such MC has transition probability matrix P:= (D^out)^-1 A [0, 1]^6 times 6, where D^out = diag(d_1^out, ..., d_6^out) is the out-degree matrix, and A {0, 1}^6 times 6 is the graph's adjacency matrix. Determine the PageRank of the Web pages in Fig. 2 by computing the stationary distribution pi of the associated MC, i.e., use e.g., Matlab to solve the linear system of equations pi = P^T pi, 1^T pi = 1. Do you find any discrepancy with your answer to A? Properties of transition probability matrices (Markov matrix). A square matrix P is called Markov (or right stochastic) if all the elements of P are non-negative; and each row sums up to 1, i.e., P1 = 1, where 1 is the column vector of all ones. Show that the transition probability matrix P:= (D^out)^-1 A of a MC is Markov. Prove that for any eigenvalue lambda of a Markov matrix, |lambda| lessthanorequalto 1. Matrix power using spectral decomposition. Show that the n-th power of a transition probability matrix P is given by P^n = V Lambda^n V^-1 where Lambda = diag(lambda_1, lambda_2, ...) is a diagonal matrix collecting the eigenvalues of P, and V is a matrix with the corresponding eigenvectors. Note this result is actually true for any diagonalizable matrix. Perron-Frobenius and limiting properties of irreducible Markov matrices. Suppose that a matrix P R_+^n times n with non-negative entries is also irreducible, meaning that some power P^k has all strictly positive entries. Under this assumption the Perron-Frobrenius theorem asserts that: P has a real eigenvalue lambda_max R whose absolute value is largest among all eigenvalues. lambda_max corresponds to a unique eigenvector v_max, with real, non-negative entries that sum to 1. Use statement 1) and part C) to show that for an irreducible Markov matrix lambda_max = 1, and that all other eigenvalues satisfy |lambda|