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# The Effect of New Links on Google Pagerank - North Carolina State ...

The Effect of New Links on Google Pagerank - North Carolina State ...

## 324 Avrachenkov and

324 Avrachenkov and LitvakDownloaded By: [North Carolina State University] At: 16:47 3 April 2007ong>Theong>n, premultiplying the above equation by 1−cn1T and using (3), we getcu T [I − cP ] −1˜ = + 1,1 − cu T [I − cP ] −1 e 1and consequently,˜ 1 = 111 − cu T [I − cP ] −1 e 1, (11)cu T [I − cP ] −1 e j˜ j = j + 1, j = 2, , n (12)1 − cu T [I − cP ] −1 e 1Next, we evaluate cu T [I − cP ] −1 e j for j = 1, , n,cu T [I − cP ] −1 e j = cu T Ze j = c k ( k1 11 +1∑)e Ti− p T 1Ze jk k 1i=2= k ( k1 c1 +1∑)e TiZe j − cp T 1k k Ze j 1i=2Since cPZ = Z − I , we have cp1 T Z = zT 1 − e 1 T , where zT 1matrix Z , and hence cp1 T Ze j = z 1j − e1 T e j. Thus, we getcu T [I − cP ] −1 e j = k ( k1 c1 +1∑)z ij − (z 1j − e T 1k k e j) 1i=2is the 1st row ong>ofong> theSubstituting the above expression for cu T [I − cP ] −1 e j , j = 1, , n, into(11) and (12), we obtain (9) and (10). □ong>Theong> results in ong>Theong>orem 3.1 are in line with formula (5). If page 1updates its outgoing links then in decomposition (5) for 1 only the secondmultiplier will be affected. In the new situation, the probability ˜q 11 to returnto page 1 starting from this page, is given bySubstituting this expression in˜q 11 = k − k 1q 11 + c k 1 +1∑q i1 k ki=2˜ 1 = ˜z 11z 11 1 = 1 − q 111 −˜q 11 1 ,

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