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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 ,

ong>Effectong> ong>ofong> ong>Newong> ong>Linksong> on Google PageRank 325Downloaded By: [North Carolina State University] At: 16:47 3 April 2007we get the updating formula (9). According to (9) the ranking ong>ofong> page 1increases when1 + c k 1 +1∑z i1 − z 11 > 0, (13)k 1which is equivalent toi=2∑q i1 > q 11 k11 +1k 1i=2Hence, the page 1 increases its ranking when it refers to pages that arecharacterized by a high value ong>ofong> q i1 . ong>Theong>se must be the pages that referto page 1 or at least belong to the same Web community. Here by a Webcommunity we mean a set ong>ofong> Web pages that a surfer can reach from oneto another in a relatively small number ong>ofong> steps.Let us now consider formula (10). First, we see that the differencebetween the old and the new ranking ong>ofong> page j is proportional to 1 .Naturally, hyperlink references from pages with high ranking have agreater impact on other pages. Furthermore, the PageRank ong>ofong> page jincreases ifkc1 +1∑z ij > z 1j (14)k 1i=2Indeed, if (14) holds then the increase ong>ofong> PageRank for page j follows from(6) since z kj increases for each page k that has a path to j via page 1, andthe other z kj ’s remain unaffected. Naturally, it is most beneficial for pagej to receive one ong>ofong> the new links. Formally, it follows from (7) that z ij =q ij z jj where q ij < 1, so that z jj constitutes the maximal possible contributionin the left-hand side ong>ofong> (14). On the other hand, if several new links areadded then the PageRank ong>ofong> page j might actually decrease even if thispage receives one ong>ofong> the new links. Such situation occurs when most ong>ofong>newly created links point to “irrelevant” pages. For instance, let j = 2 andassume that there is no hyperlink path from pages 3, , k + 1 to page 2.ong>Theong>n z ij is close to zero for i = 3, , k + 1, and the PageRank ong>ofong> page 2 willincrease only if (c/k 1 )z 22 > z 12 , which is not necessarily true, especially if z 12and k 1 are considerably large. ong>Theong> asymptotic analysis in the next sectionallows us to further clarify this issue.4. ASYMPTOTIC ANALYSISLet us apply an asymptotic analysis to formulae (9) and (10) when cis close to one and the Markov chain induced by the hyperlink matrix P

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