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1. 1. Link Analysis Rajendra AkerkarVestlandsforsking, Norway
2. 2. Objectives  To review common approaches to l k analysis h link l  To calculate the popularity of a site based on link analysis  To model human judgments indirectly2 R. Akerkar
3. 3. Outline 1. 1 Early Approaches to Link Analysis 2. Hubs and Authorities: HITS 3. Page Rank 4. Stability 5. Probabilistic Link Analysis 6. Limitation of Link Analysis3 R. Akerkar
4. 4. Early Approaches Basic Assumptions • Hyperlinks contain information about the human judgment of a site • The more incoming links to a site the more it is site, judged important Bray 1996 y • The visibility of a site is measured by the number of other sites pointing to it • The luminosity of a site is measured by the number of other sites to which it points  Limitation: failure to capture the relative importance of different parents (children) sites4 R. Akerkar
5. 5. Early Approaches Mark 1988 • To calculate the score S of a document at vertex v 1 S(v) = s(v) + Σ S(w) | ch[v] | w Є |ch(v)| v: a vertex i the h in h hypertext graph G = (V E) h (V, S(v): the global score s(v): the score if the document is isolated ch(v): children of the document at vertex v • Limitation: - Require G to be a directed acyclic g p ( q y graph (DAG) ) - If v has a single link to w, S(v) > S(w) - If v has a long path to w and s(v) < s(w), then S(v) > S (w)  unreasonable5 R. Akerkar
6. 6. Early Approaches Marchiori (1997) • Hyper information should complement textual information to obtain the overall information S(v) = s(v) + h(v) ( ) ( ) ( ) - S(v): overall information - s(v): textual information - h(v): hyper information r(v, w) • h(v) = Σ F S(w) w Є |ch[v]| - F a fading constant F Є (0 1) F: constant, (0, - r(v, w): the rank of w after sorting the children of v by S(w)  a remedy of the previous approach (Mark 1988)6 R. Akerkar
7. 7. HITS - Kleinberg’s Algorithm • HITS – Hypertext Induced Topic Selection yp p • For each vertex v Є V in a subgraph of interest: a(v) - the authority of v h(v) - the hubness of v • A site is very a thoritati e if it recei es man er authoritative receives many citations. Citation from important sites weight more than citations from less-important sites • Hubness shows the importance of a site. A good hub is a site that links to many authoritative sites y7 R. Akerkar
8. 8. Authority and Hubness 2 5 3 1 1 64 7 a(1) = h(2) + h(3) + h(4) h(1) = a(5) + a(6) + a(7)8 R. Akerkar
9. 9. Authority d H b A th it and Hubness Convergence C g • R Recursive d i dependency: d a(v)  Σ h(w) w Є pa[v] h(v)  Σ w Є ch[v] a(w) • Using Linear Algebra, we can prove: a(v) and h(v) converge9 R. Akerkar
10. 10. HITS Example Find a base subgraph:• Start with a root set R {1, 2, 3, 4}• {1, 2, 3, 4} - nodes relevant to the topic• Expand the root set R to includeall the children and a fixednumber of parents of nodes in R A new set S (base subgraph) 10 R. Akerkar
11. 11. HITS Example BaseSubgraph( R d) R, 1. S  r 2. for each v in R 3. do S  S U ch[v] 4. P  pa[v] 5. if |P| > d f 6. then P  arbitrary subset of P having size d 7. SSUP 8. 8 return S11 R. Akerkar
12. 12. HITS Example Hubs and authorities: two n-dimensional a and h HubsAuthorities(G) |V| 1 1  [1,…,1] Є R 2 a0  h 0  1 3 t 1 4 repeat 5 for each v in V 6 do at (v)  Σ h (w) w Є pa[v] t -1 7 ht (v)  Σ w Є pa[v] a (w) 8 a t  at / || at || t -1 9 h t  ht / || ht || 10 t  t+1 11 until || a t – at -1 || + || h t – ht -1 || < ε 1 1 12 return (a t , h t )12 R. Akerkar
13. 13. HITS Example Results y Authority Hubness1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Authority and hubness weights13 R. Akerkar
14. 14. HITS Improvements Brarat and Henzinger (1998) • HITS problems 1) The document can contain many identical links to the same document in another host 2) Links are generated automatically (e.g. messages posted on newsgroups) • Solutions 1) Assign weight to identical multiple edges, which are inversely proportional to their multiplicity 2) Prune irrelevant nodes or regulating the influence of a node with a relevance weight14 R. Akerkar
15. 15. PageRank Introduced by Page et al ( y g (1998))  The weight is assigned by the rank of parents Difference with HITS  HITS takes Hubness & Authority weights  The page rank is proportional to its parents’ rank, but inversely proportional to its parents’ outdegree 15
16. 16. Matrix Notation Adjacent M i Adj Matrix A= * http://www.kusatro.kyoto-u.com 16
17. 17. Matrix Notation Matrix Notation r=αBr=Mr α : eigenvalue r : eigenvector of B Ax=λx B= | A - λI | x = 0 Finding Pagerank  to find eigenvector of B with an associated eigenvalue α 17
18. 18. Matrix Notation PageRank: eigenvector of P relative to max eigenvalue B = P D P-1 D: diagonal matrix of eigenvalues {λ1, … λn} P: regular matrix that consists of eigenvectors PageRank r1 = normalized18 R. Akerkar
19. 19. Matrix Notation • Confirm the result # of inlinks from high ranked page hard to explain about 5&2 6&7 5&2, • Interesting Topic How do you create your homepage highly ranked?19 R. Akerkar
20. 20. Markov Chain Notation  Random surfer model  Description of a random walk through the Web graph  Interpreted as a transition matrix with asymptotic probability that a surfer is currently browsing that page rt = M rtt-1 1 M: transition matrix for a first-order Markov chain (stochastic) Does it converge to some sensible solution ( too) g (as ) regardless of the initial ranks ?20 R. Akerkar
21. 21. Problem  “Rank Sink” Problem  In general, many Web pages have no inlinks/outlinks  It results in dangling edges in the graph E.g. no parent  rank 0 MT converges to a matrix whose last column is all zero no children  no solution MT converges to zero matrix21 R. Akerkar
22. 22. Modification  Surfer will restart browsing by p g y picking a new Web p g at g page random M=(B+E) E : escape matrix M : stochastic matrix  S ll problem? Still bl  It is not guaranteed that M is primitive  If M is stochastic and primitive, PageRank converges to corresponding stationary distribution of M22 R. Akerkar
23. 23. PageRank Algorithm * Page et al 1998 al,23 R. Akerkar
24. 24. Distribution of the Mixture Model  The probability distribution that results from combining the p y g Markovian random walk distribution & the static rank source distribution r = εe + (1- ε)x 1 ε: probability of selecting non-linked page PageRank Now, transition matrix [εH + (1- ε)M] is primitive and stochastic rt converges to the dominant eigenvector24 R. Akerkar
25. 25. Stability  Whether the link analysis algorithms based on eigenvectors are stable in the sense that results don’t change significantly?  Th connectivity of a portion of the graph is changed The f f h h h d arbitrary  How will it affect the results of algorithms?25 R. Akerkar
26. 26. Stability of HITS Ng et al (2001) • A bound on the number of hyperlinks k that can added or deleted from one page without affecting the authority or hubness weights • It is possible to perturb a symmetric matrix by a quantity that grows as δ that produces a constant perturbation of the dominant eigenvector δ: eigengap λ1 – λ2 g g p d: maximum outdegree of G26 R. Akerkar
27. 27. Stability of PageRank Ng et al (2001) V: the set of vertices touched by the perturbation The parameter ε of the mixture model has a stabilization role If the set of pages affected by the p pg y perturbation have a small rank, the overall , change will also be small tighter bound by Bianchini et al (2001) δ(j) >= 2 depends on the edges incident on j27 R. Akerkar
28. 28. SALSA  SALSA (Lempel, Moran 2001)  Probabilistic extension of the HITS algorithm  Random walk is carried out by following hyperlinks both in the forward and in the backward direction  Two separate random walks  Hub walk  Authority walk28 R. Akerkar
29. 29. Forming a Bipartite Graph in SALSA29 R. Akerkar
30. 30. Random Walks  Hub walk  Follow a Web link from a page uh to a page wa (a forward link) and then  Immediately traverse a backlink going from wa to vh, where (u w) (u,w) Є E and (v,w) Є E  Authority Walk y  Follow a Web link from a page w(a) to a page u(h) (a backward link) and then  Immediately traverse a f d l forward link going b k f dl k back from vh to wa where (u,w) Є E and (v,w) Є E30 R. Akerkar
31. 31. Computing Weights  Hub weight computed from the sum of the product of the inverse degree of the in-links and the out-links31 R. Akerkar
32. 32. Why We Care  Lempel and Moran (2001) showed theoretically that SALSA weights are more robust that HITS weights in the presence of the Tightly Knit Community (TKC) Effect.  This effect occurs when a small collection of pages (related to a given topic) is connected so that every hub links to every authority and includes as a special case the mutual reinforcement effect  The pages in a community connected in this way can be ranked highly by HITS, higher than pages in a much larger collection HITS where only some hubs link to some authorities  TKC could be exploited by spammers hoping to increase their page weight ( li k f i h (e.g. link farms) )32 R. Akerkar
33. 33. A Similar Approach  Rafiei and Mendelzon (2000) and Ng et al. (2001) propose similar approaches using reset as in PageRank  Unlike PageRank, in this model the surfer will follow a forward link on odd steps but a backward li k b k d link on even steps t  The stability properties of these ranking distributions are similar to those of PageRank (Ng et al. 2001)33 R. Akerkar
34. 34. Overcoming TKC  Similarity downweight sequencing and sequential y g g clustering (Roberts and Rosenthal 2003)  Consider the underlying structure of clusters y g  Suggest downweight sequencing to avoid the Tight Knit Community problem g yp  Results indicate approach is effective for few tested queries, but still untested on a large scale34 R. Akerkar
35. 35. PHITS and More  PHITS: Cohn and Chang (2000)  Only the principal eigenvector is extracted using SALSA, so the authority along the remaining eigenvectors is completely neglected  Account for more eigenvectors of the co-citation matrix  See also Lempel, Moran (2003)35 R. Akerkar
36. 36. Limits of Link Analysis  META tags/ invisible text  Search engines relying on meta tags in documents are often misled (intentionally) by web developers P f Pay-for-place l  Search engine bias : organizations pay search engines and page rank  Advertisements: organizations pay high ranking pages for advertising space  W h a primary effect of increased visibility to end users and a secondary With ff f d bl d d d effect of increased respectability due to relevance to high ranking page36 R. Akerkar
37. 37. Limits of Link Analysis  Stability  Adding even a small number of nodes/edges to the graph has a significant impact  T i drift – similar t TKC Topic d ift i il to  A top authority may be a hub of pages on a different topic resulting in increased rank of the authority p g g y page  Content evolution  Adding/removing links/content can affect the intuitive authority rank of a page requiring recalculation of page ranks37 R. Akerkar
38. 38. Further Reading  R. Akerkar d P. Lingras, Building an I ll R Ak k and P L B ld Intelligent W b Th Web: Theory and Practice, Jones & Bartlett, 2008  http://www.jbpub.com/catalog/9780763741372/ p j p g38 R. Akerkar