Connected Components, and Pagerank

Transcription

1 COMP4650 Connected Components, and Pagerank Lexing Xie Research School of Computer Science, ANU Lecture slides credit: Lada Adamic, U Michigan Jure Leskovec, Stanford, Andreas Haeberlen, UPenn

2 Connected components Strongly connected components Each node within the component can be reached from every other node in the component by following directed links n Strongly connected components n B C D E n A n G H n F n Weakly connected components: every node can be reached from every other node by following links in either direcron A B E D C F H G n Weakly connected components n A B C D E n G H F A B C F G n In undirected networks one talks simply about connected components by Lada Adamic, U Michigan E D H

3 Strongly Connected Component (SCC)

4 Proof (*)

5 Giant component if the largest component encompasses a significant fracron of the graph, it is called the giant component by Lada Adamic, U Michigan

6 Why not? (*)

7 Bow- Tie Structure of the Web Andrei Broder, Ravi Kumar, Farzin Maghoul, Prabhakar Raghavan, Sridhar Rajagopalan, Raymie Stata, Andrew Tomkins, and Janet Wiener Graph structure in the Web. Comput. Netw. 33, - 6 (June 2000),

8 Not Everyone Asks/Replies The Web is a bow Re The Java Forum network is an uneven bow Re Core: A strongly connected component, in which everyone asks and answers IN: Mostly askers. OUT: Mostly Helpers Jun Zhang, Mark S. Ackerman, and Lada Adamic ExperRse networks in online communires: structure and algorithms. In Proceedings of the 6th internaronal conference on World Wide Web (WWW '07).,

9 Minkyoung Kim, Lexing Xie, Peter Christen, Event Diffusion Paherns in Social Media (202) Intl. Conf. on Weblogs and Social Media (ICWSM), 8 pages, Dublin, Ireland, June 202

10 Modern InformaRon Retrieval: The Concepts and Technology behind Search Ricardo Baeza- Yates, Berthier Ribeiro- Neto Addison- Wesley Professional; 2 ediron (February 0, 20)

11 Google's PageRank (Brin/Page 98) A technique for esrmarng page quality Based on web link graph Results are combined with IR score Think of it as: TotalScore = IR score * PageRank In pracrce, search engines use many other factors (for example, Google says it uses more than 200) A. Haeberlen, Z. Ives, U Penn

12 Shouldn't E's vote be worth more than F's? PageRank: IntuiRon G A H E B How many levels should we consider? I J F C D Imagine a contest for The Web's Best Page IniRally, each page has one vote Each page votes for all the pages it has a link to To ensure fairness, pages vorng for more than one page must split their vote equally between them VoRng proceeds in rounds; in each round, each page has the number of votes it received in the previous round In pracrce, it's a lihle more complicated - but not much! 2

13 PageRank Each page i is given a rank x i Goal: Assign the x i such that the rank of each page is governed by the ranks of the pages linking to it: xi = x j N j B i j 3 Rank of page i How do we compute the rank values? Every page j that links to i Rank of page j Number of links out from page j

14 IteraRve PageRank (simplified) Initialize all ranks to be equal, e.g.: (0) x i = n Iterate until convergence x ( k+ ) = x ( k ) i j N j B j i 4

15 Example: Step 0 Initialize all ranks to be equal (0) x i = n

16 Example: Step Propagate weights across out-edges x = x ( k+ ) ( k ) i j j B N j i

17 Example: Step 2 Compute weights based on in-edges ( ) (0) 0.50 x i = j B i N j x j

18 Example: Convergence x = x ( k+ ) ( k ) i j j B N j i

19 Naïve PageRank Algorithm Restated Let N(p) = number outgoing links from page p B(p) = number of back- links to page p PageRank( p) = PageRank( b) N( b) Each page b distributes its importance to all of the pages it points to (so we scale by /N(b)) b B p Page p s importance is increased by the importance of its back set 9

20 In Linear Algebra formularon Create an m x m matrix M to capture links: M(i, j) = / n j if page i is pointed to by page j and page j has n j outgoing links = 0 otherwise IniRalize all PageRanks to, mulrply by M repeatedly unrl all values converge: PageRank ( p PageRank ( p... PageRank ( p 2 ') ') ') ) ) ) Computes principal eigenvector via power iteraron = PageRank ( p PageRank ( p M... PageRank ( m p m 2 20

21 A Brief Example Google Amazon Yahoo g' y a = * g y a Running for multiple iterations: g y a =, 0.5.5, , Total rank sums to number of pages 2

22 Oops # PageRank Sinks Google Amazon Yahoo g' y a = * g y a Running for multiple iterations: 'dead end' - PageRank is lost auer each round g y a =, , ,,

23 Oops #2 PageRank hogs Google Amazon Yahoo g' y a = * g y a Running for multiple iterations: PageRank cannot flow out and accumulates g y a =, , ,,

24 Improved PageRank Remove out- degree 0 nodes (or consider them to refer back to referrer) Add decay factor d to deal with sinks PageRank ( p) = ( d) + d PageRank ( b) N( b) Typical value: d=0.85 b B p 24

25 Stopping the Hog Google Amazon Yahoo g' y a = 0.85 * g y a Running for multiple iterations: g y = ,,,,, a though does this seem right? 25

26 Random Surfer Model PageRank has an intuirve basis in random walks on graphs Imagine a random surfer, who starts on a random page and, in each step, with probability d, klicks on a random link on the page with probability - d, jumps to a random page (bored?) The PageRank of a page can be interpreted as the fracron of steps the surfer spends on the corresponding page TransiRon matrix can be interpreted as a Markov Chain 26

27 PageRank, random walk on graphs G W = (- d)*g + d*u + Pagerank vector with probability d, klicks on a random link on the page with probability - d, jumps to a random page (bored?) TransiRon matrix W can be interpreted as a Markov Chain COMP4650 L Xie, ANU Computer Science

28 COMP4650 L Xie, ANU Computer Science 28

29 Search Engine OpRmizaRon (SEO) Has become a big business White- hat techniques Google webmaster tools Add meta tags to documents, etc. Black- hat techniques Link farms Keyword stuffing, hidden text, meta- tag stuffing,... Spamdexing IniRal soluron: <a rel="nofollow" href="...">...</a> Some people started to abuse this to improve their own rankings Doorway pages / cloaking Special pages just for search engines BMW Germany and Ricoh Germany banned in February 2006 Link buying 29

30 Recap: PageRank EsRmates absolute 'quality' or 'importance' of a given page based on inbound links Query- independent Can be computed via fixpoint iteraron Can be interpreted as the fracron of Rme a 'random surfer' would spend on the page Several refinements, e.g., to deal with sinks Considered relarvely stable But vulnerable to black- hat SEO An important factor, but not the only one Overall ranking is based on many factors (Google: >200) 30

31 What could be the other 200 factors? On- page Off- page Posi9ve Keyword in Rtle? URL? Keyword in domain name? Quality of HTML code Page freshness Rate of change... High PageRank Anchor text of inbound links Links from authority sites Links from well- known sites Domain expiraron date... Nega9ve Links to 'bad neighborhood' Keyword stuffing Over- oprmizaron Hidden content (text has same color as background) AutomaRc redirect/refresh... Fast increase in number of inbound links (link buying?) Link farming Different pages user/spider Content duplicaron... Note: This is enrrely specularve! 3 Source: Web InformaRon Systems, Prof. Beat Signer, VU Brussels

32 Summary Macro structure of networks Strongly and weakly connected components ApplicaRons in web/discussion forums etc. PageRank algorithm Computed using power iterarons Used for ranking webpages Can be thought of as random walk on graphs