## 324 Avrachenkov and

324 Avrachenkov and LitvakDownloaded By: [**North** **Carolina** **State** University] At: 16:47 3 April 2007ong>Theong>n, premultiplying the above equati**on** by 1−cn1T and using (3), we getcu T [I − cP ] −1˜ = + 1,1 − cu T [I − cP ] −1 e 1and c**on**sequently,˜ 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 expressi**on** 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 decompositi**on** (5) for 1 **on**ly the sec**on**dmultiplier will be affected. In the new situati**on**, the probability ˜q 11 to returnto page 1 starting from this page, is given bySubstituting this expressi**on** 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 bel**on**g to the same Web community. Here by a Webcommunity we mean a set ong>ofong> Web pages that a surfer can reach from **on**eto another in a relatively small number ong>ofong> steps.Let us now c**on**sider formula (10). First, we see that the differencebetween the old and the new ranking ong>ofong> page j is proporti**on**al 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 **on**e 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 c**on**stitutes the maximal possible c**on**tributi**on**in 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 **on**e ong>ofong> the new links. Such situati**on** 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 **on**ly if (c/k 1 )z 22 > z 12 , which is not necessarily true, especially if z 12and k 1 are c**on**siderably large. ong>Theong> asymptotic analysis in the next secti**on**allows us to further clarify this issue.4. ASYMPTOTIC ANALYSISLet us apply an asymptotic analysis to formulae (9) and (10) when cis close to **on**e and the Markov chain induced by the hyperlink matrix P