1 CSCI 5417 Information Retrieval Systems Jim Martin!Lecture 20 11/3/2011 Today  Finish PageRank  HITs  Start ML-based ranking2 11/11/11 CSCI 5417 - IR 3 PageRank Sketch  The pagerank of a page is based on the pagerank of the pages that point at it.  Roughly € Pr(P) =Pr(in)V (in)in∈P∑11/11/11 CSCI 5417 - IR 4 PageRank scoring  Imagine a browser doing a random walk on web pages:  Start at a random page  At each step, go out of the current page along one of the links on that page, equiprobably  “In the steady state” each page has a long-term visit rate - use this as the page’s score  Pages with low rank are pages rarely visited during a random walk 1/3 1/3 1/33 11/11/11 CSCI 5417 - IR 5 Not quite enough  The web is full of dead-ends. Pages that are pointed to but have no outgoing links  Random walk can get stuck in such dead-ends  Makes no sense to talk about long-term visit rates in the presence of dead-ends. ?? 11/11/11 CSCI 5417 - IR 6 Teleporting  At a dead end, jump to a random web page  At any non-dead end, with probability 10%, jump to a random web page  With remaining probability (90%), go out on a random link.  10% - a parameter (call it alpha)4 11/11/11 CSCI 5417 - IR 7 Result of teleporting  Now you can’t get stuck locally.  There is a long-term rate at which any page is visited  How do we compute this visit rate?  Can’t directly use the random walk metaphor State Transition Probabilities We’re going to use the notion of a transition probability. If we’re in some particular state, what is the probability of going to some other particular state from there. If there are n states (pages) then we need an n x n table of probabilities. 11/11/11 CSCI 5417 - IR 85 Markov Chains  So if I’m in a particular state (say the start of a random walk)  And I know the whole n x n table  Then I can compute the probability distribution over all the next states I might be in in the next step of the walk...  And in the step after that  And the step after that 11/11/11 CSCI 5417 - IR 9  Say alpha = .5 Example 11/11/11 CSCI 5417 - IR 10 1 3 26  Say alpha = .5 Example 11/11/11 CSCI 5417 - IR 11 1 3 2 ? P(32)  Say alpha = .5 Example 11/11/11 CSCI 5417 - IR 12 1 3 2 2/3 P(32)7  Say alpha = .5 Example 11/11/11 CSCI 5417 - IR 13 1 3 2 1/6 2/3 1/6 P(3*)  Say alpha = .5 Example 11/11/11 CSCI 5417 - IR 14 1 3 2 1/6 2/3 1/6 5/12 1/6 5/12 1/6 2/3 1/68  Say alpha = .5 Example 11/11/11 CSCI 5417 - IR 15 1 3 2 1/6 2/3 1/6 5/12 1/6 5/12 1/6 2/3 1/6 Assume we start a walk in 1 at time T0. Then what should we believe about the state of affairs in T1? What should we believe about things at T2? Example 11/11/11 CSCI 5417 - IR 16 PageRank values9 17#More Formally  A probability (row) vector x = (x1 , ..., xN) tells us where the random walk is at any point.  Example:  More generally: the random walk is on the page i with probability xi.  Example:  Σ xi = 1"( 0 0 0 … 1 … 0 0 0 ) 1 2 3 … i … N-2 N-1 N ( 0.05 0.01 0.0 … 0.2 … 0.01 0.05 0.03 ) 1 2 3 … i … N-2 N-1 N 18#Change##in##probability##vector# If#the#probability#vector#is#x"= (x1 , ..., xN),"at#this#step,#what#is#it#at#the#next#step?#10 19#Change##in##probability##vector# If#the#probability#vector#is#x"="(x1 , ..., xN),"at#this#step,#what#is#it#at#the#next#step?# Recall#that#row#i"#of#the#transi@on#probability#matrix#P"tells#us#where#we#go#next#from#state#i.#20#Change##in##probability##vector# If#the#probability#vector#is#x"="(x1","...,"xN),"at#this#step,#what#is#it#at#the#next#step?# Recall#that#row#i"#of#the#transi@on#probability#matrix#P"tells#us#where#we#go#next#from#state#i." So#from#x,#our#next#state#is#distributed#as#xP.#11 21#Steady#state#in#vector#nota@on#22#Steady#state#in#vector#nota@on# The#steady#state#in#vector#nota@on#is#simply#a#vector#"""""π#=#(π1,#π2,#…,#πΝ)#of#probabili@es.#12 23#Steady#state#in#vector#nota@on# The#steady#state#in#vector#nota@on#is#simply#a#vector#"""""π#=#(π1,#π2,#…,#πΝ)#of#probabili@es.# Use#π#to#dis@nguish#it#from#the#nota@on#for#the#probability#vector#x.)"24#Steady#state#in#vector#nota@on# The#steady#state#in#vector#nota@on#is#simply#a#vector#"""""π#=#(π1,#π2,#…,#πΝ)#of#probabili@es.# Use#π#to#dis@nguish#it#from#the#nota@on#for#the#probability#vector#x.)" π#is#the#longNterm#visit#rate#(or#PageRank)#of#page#i.#13 25#Steady#state#in#vector#nota@on# The#steady#state#in#vector#nota@on#is#simply#a#vector#"""""π#=#(π1,#π2,#…,#πΝ)#of#probabili@es.# Use#π#to#dis@nguish#it#from#the#nota@on#for#the#probability#vector#x.)" π#is#the#longNterm#visit#rate#(or#PageRank)#of#page#i.# So#we#can#think#of#PageRank#as#a#very#long#vector#–#one#entry#per#page.#26#SteadyNstate#distribu@on:#Example#14 27#SteadyNstate#distribu@on:#Example# What#is#the#PageRank#/#steady#state#in#this#example?#28#SteadyNstate#distribu@on:#Example#15 29#SteadyNstate#distribu@on:#Example#x1 Pt(d1) x2 Pt(d2) P11 = 0.25 P21 = 0.25 P12 = 0.75 P22 = 0.75 t0 t1 0.25 0.75 Pt(d1) = Pt-1(d1) * P11 + Pt-1(d2) * P21 Pt(d2) = Pt-1(d1) * P12 + Pt-1(d2) * P22 30#SteadyNstate#distribu@on:#Example#x1 Pt(d1) x2 Pt(d2) P11 = 0.25 P21 = 0.25 P12 = 0.75 P22 = 0.75 t0 t1 0.25 0.75 0.25 0.7516 31#SteadyNstate#distribu@on:#Example#x1 Pt(d1) x2 Pt(d2) P11 = 0.25 P21 = 0.25 P12 = 0.75 P22 = 0.75 t0 t1 0.25 0.25 0.75 0.75 0.25 0.75 PageRank vector = π = (π1,#π2) = (0.25, 0.75) Pt(d1) = Pt-1(d1) * P11 + Pt-1(d2) * P21 Pt(d2) = Pt-1(d1) * P12 + Pt-1(d2) * P22 32#SteadyNstate#distribu@on:#Example#x1 Pt(d1) x2 Pt(d2) P11 = 0.25 P21 = 0.25 P12 = 0.75 P22 = 0.75 t0 t1 0.25 0.25 0.75 0.75 0.25 0.75 (convergence) PageRank vector = π = (π1,#π2) = (0.25, 0.75) Pt(d1) = Pt-1(d1) * P11 + Pt-1(d2) * P21 Pt(d2) = Pt-1(d1) * P12 + Pt-1(d2) * P2217 33#How#do#we#compute#the#steady#state#vector?#34#How#do#we#compute#the#steady#state#vector?# In#other#words:#how#do#we#compute#PageRank?#18 35#How#do#we#compute#the#steady#state#vector?# In#other#words:#how#do#we#compute#PageRank?# Recall:#π#=#(π1,#π2,#…,#πN)#is#the#PageRank##vector,#the#vector#of#steadyNstate#probabili@es#...#36#How#do#we#compute#the#steady#state#vector?#