R i is a vector with n entries r s 1s i 2 s r i why

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r i  is a vector with N entries r S  = (1/|S|)  i 2 S   r i   Why is linearity important? Instead of 2 N  biased page rank vectors we need to  store N vectors
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Linearity example  1 2 3 4 5 Let us compute r {1,2}  for  β  = 0.8 1 2 3 4 5 0.4 0.4 0.8 0.8 0.4 0.4 0.8 0.1 0.1 Node Iteration 0 1 2… stable 1 0.1 0.1 0.164 0.300 2 0.1 0.14 0.172 0.323 3 0 0.04 0.04 0.120 4 0 0.04 0.056 0.130 5 0 0.04 0.056 0.130
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Linearity example 0.407 0.239 0.163 0.096 0.096 0.192 0.407 0.077 0.163 0.163 0.300 0.323 0.120 0.130 0.130 1 2 3 4 5 0.300 0.323 0.120 0.130 0.130 r {1,2} r 1 r 2 (r 1 +r 2 )/2
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Intuition behind proof Let’s use the many-random-walkers model  with  M  random walkers Let us color a random walker with color  i  if his  most recent teleport was to page  i At time t, we expect  M/|S|  of the random  walkers to be colored i At any page  j , we would therefore expect to  find  (M/|S|)r i (j)  random walkers colored i So total number of random walkers at page  j  =  (M/|S|) i 2 S r i (j)
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Basis Vectors Suppose T = union of all teleport sets of interest Call it the teleport universe We can compute the rank vector corresponding to any  teleport set S µ T as a linear combination of the vectors   r i  for i 2 We call these vectors the  basis vectors  for T We can also compute rank vectors where we assign  different weights to teleport pages
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Decomposition Still too many basis vectors E.g., |T| might be in the thousands N|T| values Decompose basis vectors into  partial vectors   and  hubs skeleton
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Tours Consider a random walker with teleport set {i} Suppose walker is currently at node j  The random walker’s  tour  is the sequence of nodes on  the walker’s path since the last teleport E.g., i,a,b,c,a,j Nodes can repeat in tours – why? Interior nodes  of the tour = {a,b,c} Start node  = {i},  end node  = {j} A page can be both start node and interior node, etc
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Tour splitting Consider random walker with teleport set {i},  biased rank vector r i r i (j) = probability random walker reaches j by  following some tour with start node i and end  node j Consider node k  Can have k = j but not k = i i k j
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Tour splitting Let  r i k (j)  be the probability that random surfer reaches  page j through a tour that  includes  page k as an  interior or end node. Let  r i ~k (j)  be the probability that random surfer reaches  page j through a tour that  does not  include k as an  interior or end node. r i (j) = r i k (j) + r i ~k (j)   i k j
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Example 1 2 3 4 5 0.4 0.4 0.8 0.8 0.4 0.4 0.8 0.2 Let us compute r 1 ~2  for  β  = 0.8 Node Iteration 0 1 2… stable 1 0.2 0.2 0.264 0.294 2 0 0 0 0 3 0 0.08 0.08 0.118 4 0 0 0 0 5 0 0 0 0 Note that many entries are  zeros 1 2 3 4 5 0.4 0.4 0.8 0.8 0.4 0.4 0.8 0.2
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Example 1 2 3 4 5 0.4 0.4 0.8 0.8 0.4 0.4 0.8 0.2 Let us compute r 2 ~2  for  β  = 0.8 Node Iteration 0 1 2… stable 1 0 0 0.064 0.094 2 0.2 0.2 0.2 0.2 3 0 0 0 0.038 4 0 0.08 0.08 0.08 5 0 0.08 0.08 0.08
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Rank composition Notice: r 1 2
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