## Question

###### 1.2.4) Limiting PageRank values [12 points]. The directedl graph below depicts the hyperlinks among six web pages {A_ B.C.D.E. F} along with proposed PageRank centrality values for each Web Wag. In ts exercise assumn teleportation evaporation-Tain component4) {2 points} Are these correct (limniting) equilibriu vales lor the basic PageRank update rule? poinlsh PageRank values correspond t the stationary tlistribution Markov chain (MC), which dleseribes the secpuence ol page visits ol random weh s

1.2.4) Limiting PageRank values [12 points]. The directedl graph below depicts the hyperlinks among six web pages {A_ B.C.D.E. F} along with proposed PageRank centrality values for each Web Wag. In ts exercise assumn teleportation evaporation-Tain component 4) {2 points} Are these correct (limniting) equilibriu vales lor the basic PageRank update rule? poinlsh PageRank values correspond t the stationary tlistribution Markov chain (MC), which dleseribes the secpuence ol page visits ol random weh sucfer. Such MC has trausition prohability matrix (Damt ) - [0, 1]6*6 . whcre Domt dliag(dqut dgut is the out-degree matrix, Andl {0,1}6*6 is the graph $ axljacency matrix Determine the PageRank of the wel pages hy computing the stationaly distribution of thc associated MC. iC.1 solve the following set of cqpuations (YOn Can Use aWY programming language) T =P 1Tw =1. Is this consistent with Four answer to a)? c) /8 points square matrix is called Markov (or right stochastic} if i} all the elements of JOI- negative: and ii) each IOW SUMS "p 10 Show that the transition probability matrix P := (Dout ) MC' is Markov . Prove that for aHy eigenvale Markov matrix |Xl < 4 (Hint: check Gershgorin circle theorem) _ points} Suppose that matrix P is also irreducible Aud "periodic (see Perron-Frobenius theoremn) . Show that lit il t for all x e R" whcrc constant that depends on IF P is the transition prohability matrix of an irreduciblc andd aperiodic MC, what the rclationship betwecn And the stationary dlistribution Motivatecl by you ASWCT Call VOU specily simple iterative procedure to obtain the PageRank vector? Implement such pIO cexlure for OUI network of interest and illustrate the speed of convergence with plot.