The PageRank Citation Ranking:The PageRank Citation Ranking:Bringing Order to the WebBringing Order to the Web

Larry Page etc.

Stanford University

Presented by

Guoqiang Su & Wei Li

ContentsContents

Motivation Related work Page Rank & Random Surfer Model Implementation Application Conclusion

MotivationMotivation

Web: heterogeneous and unstructured Free of quality control on the web Commercial interest to manipulate ranking

Related WorkRelated Work

Academic citation analysis Link-based analysis Clustering methods of link structure Hubs & Authorities Model

BacklinkBacklink

Link Structure of the Web Approximation of importance / quality

PageRankPageRank

Pages with lots of backlinks are important Backlinks coming from important pages

convey more importance to a page

Problem: Rank Sink

∑∈

=uBv vN

vRcuR

)()(

Rank SinkRank Sink Page cycles pointed by some incoming link

Problem: this loop will accumulate rank but never distribute any rank outside

Escape TermEscape Term

Solution: Rank Source

c is maximized and = 1 E(u) is some vector over the web pages

– uniform, favorite page etc.

)()(

)( ucEN

vRcuR

uBv v

+= ∑∈

1R

Matrix NotationMatrix Notation

R is the dominant eigenvector and c is the dominant eigenvalue of because c is maximized

ReEAcR TT )( ×+=

)( TeEA ×+

Computing PageRankComputing PageRank

- initialize vector over web pages

loop:

- new ranks sum of normalized backlink ranks

- compute normalizing factor

- add escape term

- control parameter

while - stop when converged

SR ←0

iT

i RAR ←+1

111 +−← ii RRd

dERR ii +← ++ 11

ii RR −← +1σ

εσ >

Random Surfer ModelRandom Surfer Model

Page Rank corresponds to the probability distribution of a random walk on the web graphs

E(u) can be re-phrased as the random surfer gets bored periodically and jumps to a different page and not kept in a loop forever

ImplementationImplementation Computing resources — 24 million pages — 75 million URLs

Memory and disk storage

Weight Vector

(4 byte float)

Matrix A (linear access)

Implementation (Con't)Implementation (Con't)

Unique integer ID for each URL Sort and Remove dangling links Rank initial assignment Iteration until convergence Add back dangling links and Re-compute

Convergence PropertiesConvergence Properties Graph (V, E) is an expander with factor α if

for all (not too large) subsets S: |As|≥ α|s| Eigenvalue separation: Largest eigenvalue

is sufficiently larger than the second-largest eigenvalue

Random walk converges fast to a limiting probability distribution on a set of nodes in the graph.

Convergence Properties (con't)Convergence Properties (con't) PageRank computation is O(log(|V|)) due to

rapidly mixing graph G of the web.

Personalized PageRankPersonalized PageRank Rank Source E can be initialized :

– uniformly over all pages: e.g. copyright warnings, disclaimers, mailing lists archives

✦ result in overly high ranking– total weight on a single page, e.g. Netscape, McCarthy

✦ great variation of ranks under different single pages as rank source

– and everything in-between, e.g. server root pages

✦ allow manipulation by commercial interests

Applications IApplications I Estimate web traffic

– Server/page aliases

– Link/traffic disparity, e.g. porn sites, free web-mail

Backlink predictor– Citation counts have been used to predict future citations

– very difficult to map the citation structure of the web completely

– avoid the local maxima that citation counts get stuck in and get better performance

Applications II - Ranking ProxyApplications II - Ranking Proxy

Surfer's Navigation Aid

Annotating links by PageRank (bar graph)

Not query dependent

IssuesIssues Users are no random walkers – Content based methods

Starting point distribution – Actual usage data as starting vector

Reinforcing effects/bias towards main pages How about traffic to ranking pages? No query specific rank Linkage spam – PageRank favors pages that managed to get other pages to link to them – Linkage not necessarily a sign of relevancy, only of promotion (advertisement…)

Evaluation IEvaluation I

Evaluation IIEvaluation II

ConclusionConclusion PageRank is a global ranking based on the

web's graph structure PageRank use backlinks information to

bring order to the web PageRank can separate out representative

pages as cluster center A great variety of applications