Welcome back!• The final exam is on Friday, December 12, 7-10pmIf you have a conflict (overlapping exam, or more than 2 exa ms within 24h), please emailMahmood until Sunday to sign-up for the Mon day conflict.• What are th e eigenspaces of1 20 3?◦ λ = 1 has eigenspace Nul0 20 2= spann10o◦ λ = 3 has eigenspace Nul−2 20 0= spann11o◦ INCORRECT: eigen space spann10,11oTransition matricesPowers of matrices can describe tr ansition of a system.Example 1. (review)•Fibonacci numbe rs Fn: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,• Fn+1= Fn+ Fn− 1Fn +1Fn=1 11 0FnFn − 1• Hence:Fn+1Fn=1 11 0nF1F0Example 2. Consider a fixed population of people with or without a job. Suppose that,each year, 50% of those unemployed find a job while 10 % of those employed loose theirjob.What is the unempl oyment rate in the long term equilibrium?Solution.employed no job0.10.90.50.5xt: proportion of popu l ation empl oyed at time t (in years)yt: proportion of po p ulation un e mployed at time txt+1yt+1=0.9xt+ 0.5yt0.1xt+ 0.5yt=0.9 0.50.1 0.5xtytThe matrix0 .9 0 .50 .1 0 .5is a Markov matrix. Its col umns add to 1 and it has no negative entries.Armin [email protected]x∞y∞is an eq uilibrium ifx∞y∞=0.9 0.50.1 0.5x∞y∞.In other words,x∞y∞is an ei g envector with eigenvalue 1.Eigenspace ofλ = 1: Nul−0.1 0.50.1 −0.5= spann51oSince x∞+ y∞= 1, we conclude thatx∞y∞=5/61/6.Hence, the u nemployment rate in the long term equilibrium is1/6.Page rankGoogle’s success i s based on an algorithm to rank webpages, thePage rank, n a medafter Google founder Larry Page.The basic idea is to determine h ow likely it is that a web user randomly gets to a g ivenwebpage. The webpages are then ranked by these probabilities.Example 3. Suppose the intern e t consisted of o nly the four webpages A, B, C, D linkedas in the following graph:Imagine a surfer foll owing these links at random.For the probabilityPRn(A) that she is at A (after n steps), we add:• the probability that she was at B (at exactly one time step before),and left forA, (that’s PRn − 1(B) ·12)• the probability that she wa s at C, and left for A,• the probability that she wa s at D, and left for A.A BC D• Hence: PRn(A) = P Rn−1(B) ·12+ PRn− 1(C) ·11+ PRn− 1(D) ·01•PRn(A)PRn(B)PRn(C)PRn(D)=0121 0130 0 0130 0 113120 0PRn − 1(A)PRn − 1(B)PRn − 1(C)PRn − 1(D)• The PageRank vectorPR (A)PR (B)PR (C)PR (D)=PR∞(A)PR∞(B)PR∞(C)PR∞(D)is the long-term equilibrium.It is an eigenvector of the Markov matrix with eigen value1.•−1121 013−1 0 0130 −1 113120 −1>RREF1 0 0 −20 1 0 −230 0 1 −530 0 0 0eigenspace of λ = 1 spanned by223531Armin [email protected]PR (A)PR (B)PR (C)PR (D)=316223531=0.3750.1250.3130.188This the PageRank vector.• The corresponding ranking of the webpages is A, C , D, B.Practice pro blemsProblem 1. Can you see why 1 is an eigenvalu e for every Markov matrix?Problem 2. ( just for fun) The real web contains pages which ha ve no outgoing links.In that case, our ra ndom surfer would ge t “stuck” (the transition matrix is not a Mar kovmatrix). Do you have an idea how to deal w ith this issue?Armin
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