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The Web as a Directed Graph

The Web as a Directed Graph
Introduction to Information Retrieval Introduction to Information Retrieval Link analysis www.ThesisScientist.com 1Introduction to Information Retrieval Today’s lecture – hypertext and links  We look beyond the content of documents  We begin to look at the hyperlinks between them  Address questions like  Do the links represent a conferral of authority to some pages? Is this useful for ranking?  How likely is it that a page pointed to by the CERN home page is about high energy physics  Big application areas  The Web  Email  Social networks www.ThesisScientist.com 2Introduction to Information Retrieval Links are everywhere  Powerful sources of authenticity and authority  Mail spam – which email accounts are spammers?  Host quality – which hosts are “bad”?  Phone call logs  The Good, The Bad and The Unknown ? Good ? ? Bad ? www.ThesisScientist.com 3Introduction to Information Retrieval Example 1: Good/Bad/Unknown  The Good, The Bad and The Unknown  Good nodes won’t point to Bad nodes  All other combinations plausible ? Good ? ? Bad ? www.ThesisScientist.com 4Introduction to Information Retrieval Simple iterative logic  Good nodes won’t point to Bad nodes  If you point to a Bad node, you’re Bad  If a Good node points to you, you’re Good ? Good ? ? Bad ? www.ThesisScientist.com 5Introduction to Information Retrieval Simple iterative logic  Good nodes won’t point to Bad nodes  If you point to a Bad node, you’re Bad  If a Good node points to you, you’re Good Good ? Bad ? www.ThesisScientist.com 6Introduction to Information Retrieval Simple iterative logic  Good nodes won’t point to Bad nodes  If you point to a Bad node, you’re Bad  If a Good node points to you, you’re Good Good Bad www.ThesisScientist.com 7 Sometimes need probabilistic analogs – e.g., mail spamIntroduction to Information Retrieval Example 2: In-links to pages – unusual patterns  Spammers violating power laws www.ThesisScientist.com 8Introduction to Information Retrieval Many other examples of link analysis  Social networks are a rich source of grouping behavior  E.g., Shoppers’ affinity – Goel+Goldstein 2010  Consumers whose friends spend a lot, spend a lot themselves  http://www.cs.cornell.edu/home/kleinber/networks-book/ www.ThesisScientist.com 9Introduction to Information Retrieval Our primary interest in this course  Link analysis for most IR functionality thus far based purely on text  Scoring and ranking  Link-based clustering – topical structure from links  Links as features in classification – documents that link to one another are likely to be on the same subject  Crawling  Based on the links seen, where do we crawl next? www.ThesisScientist.com 10Introduction to Information Retrieval Sec. 21.1 The Web as a Directed Graph hyperlink Page B Anchor Page A Hypothesis 1: A hyperlink between pages denotes a conferral of authority (quality signal) Hypothesis 2: The text in the anchor of the hyperlink on page A describes the target page B www.ThesisScientist.com 11Introduction to Information Retrieval Assumption 1: reputed sites www.ThesisScientist.com 12Introduction to Information Retrieval Assumption 2: annotation of target www.ThesisScientist.com 13Introduction to Information Retrieval Sec. 21.1.1 Anchor Text WWW Worm - McBryan Mcbr94  For ibm how to distinguish between:  IBM’s home page (mostly graphical)  IBM’s copyright page (high term freq. for ‘ibm’)  Rival’s spam page (arbitrarily high term freq.) “IBM home page” “ibm.com” “ibm” A million pieces of anchor text with “ibm” www.ibm.com send a strong signal www.ThesisScientist.com 14Introduction to Information Retrieval Sec. 21.1.1 Indexing anchor text  When indexing a document D, include (with some weight) anchor text from links pointing to D. Armonk, NY-based computer giant IBM announced today www.ibm.com Joe’s computer hardware Big Blue today announced links record profits for the quarter Sun HP IBM www.ThesisScientist.com 15Introduction to Information Retrieval Sec. 21.1.1 Indexing anchor text  Can sometimes have unexpected effects, e.g., spam, miserable failure  Can score anchor text with weight depending on the authority of the anchor page’s website  E.g., if we were to assume that content from cnn.com or yahoo.com is authoritative, then trust (more) the anchor text from them  Increase the weight of off-site anchors (non-nepotistic scoring) www.ThesisScientist.com 16Introduction to Information Retrieval Connectivity servers Getting at all that link information Inexpensively www.ThesisScientist.com 17Introduction to Information Retrieval Sec. 20.4 Connectivity Server  Support for fast queries on the web graph  Which URLs point to a given URL?  Which URLs does a given URL point to? Stores mappings in memory from  URL to outlinks, URL to inlinks  Applications  Link analysis  Web graph analysis  Connectivity, crawl optimization  Crawl control www.ThesisScientist.com 18Introduction to Information Retrieval Sec. 20.4 Boldi and Vigna 2004  http://www2004.org/proceedings/docs/1p595.pdf  Webgraph – set of algorithms and a java implementation  Fundamental goal – maintain node adjacency lists in memory  For this, compressing the adjacency lists is the critical component www.ThesisScientist.com 19Introduction to Information Retrieval Sec. 20.4 Adjacency lists  The set of neighbors of a node  Assume each URL represented by an integer  E.g., for a 4 billion page web, need 32 bits per node  Naively, this demands 64 bits to represent each hyperlink  Boldi/Vigna get down to an average of 3 bits/link  Further work achieves 2 bits/link www.ThesisScientist.com 20Introduction to Information Retrieval Sec. 20.4 Adjaceny list compression  Properties exploited in compression:  Similarity (between lists)  Locality (many links from a page go to “nearby” pages)  Use gap encodings in sorted lists  Distribution of gap values www.ThesisScientist.com 21Introduction to Information Retrieval Sec. 20.4 Main ideas of Boldi/Vigna  Consider lexicographically ordered list of all URLs, e.g.,  www.stanford.edu/alchemy  www.stanford.edu/biology  www.stanford.edu/biology/plant  www.stanford.edu/biology/plant/copyright  www.stanford.edu/biology/plant/people  www.stanford.edu/chemistry www.ThesisScientist.com 22Introduction to Information Retrieval Sec. 20.4 Boldi/Vigna  Each of these URLs has an adjacency list Why 7?  Main idea: due to templates, the adjacency list of a node is similar to one of the 7 preceding URLs in the lexicographic ordering  Express adjacency list in terms of one of these  E.g., consider these adjacency lists  1, 2, 4, 8, 16, 32, 64  1, 4, 9, 16, 25, 36, 49, 64  1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144  1, 4, 8, 16, 25, 36, 49, 64 www.ThesisScientist.com 23 Encode as (-2), remove 9, add 8Introduction to Information Retrieval Sec. 20.4 Gap encodings  Given a sorted list of integers x, y, z, …, represent by x, y-x, z-y, …  Compress each integer using a code  code - Number of bits = 1 + 2 lg x d code: …  Information theoretic bound: 1 + lg x bits z code: Works well for integers from a power law Boldi Vigna DCC 2004 www.ThesisScientist.com 24Introduction to Information Retrieval Sec. 20.4 Main advantages of BV  Depends only on locality in a canonical ordering  Lexicographic ordering works well for the web  Adjacency queries can be answered very efficiently  To fetch out-neighbors, trace back the chain of prototypes  This chain is typically short in practice (since similarity is mostly intra-host)  Can also explicitly limit the length of the chain during encoding  Easy to implement one-pass algorithm www.ThesisScientist.com 25Introduction to Information Retrieval Link analysis: Pagerank www.ThesisScientist.com 26Introduction to Information Retrieval Citation Analysis  Citation frequency  Bibliographic coupling frequency  Articles that co-cite the same articles are related  Citation indexing  Who is this author cited by? (Garfield 1972)  Pagerank preview: Pinsker and Narin ’60s  Asked: which journals are authoritative? www.ThesisScientist.com 27Introduction to Information Retrieval The web isn’t scholarly citation  Millions of participants, each with self interests  Spamming is widespread  Once search engines began to use links for ranking (roughly 1998), link spam grew  You can join a link farm – a group of websites that heavily link to one another www.ThesisScientist.com 28Introduction to Information Retrieval Sec. 21.2 Pagerank scoring  Imagine a user doing a random walk on web pages:  Start at a random page 1/3 1/3  At each step, go out of the 1/3 current page along one of the links on that page, equiprobably “In the long run” each page has a long-term visit rate - use this as the page’s score. www.ThesisScientist.com 29Introduction to Information Retrieval Sec. 21.2 Not quite enough  The web is full of dead-ends.  Random walk can get stuck in dead-ends.  Makes no sense to talk about long-term visit rates. ?? www.ThesisScientist.com 30Introduction to Information Retrieval Sec. 21.2 Teleporting  At a dead end, jump to a random web page.  At any non-dead end, with probability 10%, jump to a random web page.  With remaining probability (90%), go out on a random link.  10% - a parameter. www.ThesisScientist.com 31Introduction to Information Retrieval Sec. 21.2 Result of teleporting  Now cannot get stuck locally.  There is a long-term rate at which any page is visited (not obvious, will show this).  How do we compute this visit rate? www.ThesisScientist.com 32Introduction to Information Retrieval Sec. 21.2.1 Markov chains  A Markov chain consists of n states, plus an nn transition probability matrix P.  At each step, we are in one of the states.  For 1  i,j  n, the matrix entry P tells us the ij probability of j being the next state, given we are currently in state i. P 0 ii is OK. i j P ij www.ThesisScientist.com 33Introduction to Information Retrieval Sec. 21.2.1 Markov chains n P1.  ij  Clearly, for all i, j1  Markov chains are abstractions of random walks.  Exercise: represent the teleporting random walk from 3 slides ago as a Markov chain, for this case: www.ThesisScientist.com 34Introduction to Information Retrieval Sec. 21.2.1 Ergodic Markov chains  For any ergodic Markov chain, there is a unique long-term visit rate for each state.  Steady-state probability distribution.  Over a long time-period, we visit each state in proportion to this rate.  It doesn’t matter where we start. www.ThesisScientist.com 35Introduction to Information Retrieval Sec. 21.2.1 Probability vectors  A probability (row) vector x = (x , … x ) tells us 1 n where the walk is at any point.  E.g., (000…1…000) means we’re in state i. 1 i n More generally, the vector x = (x , … x ) 1 n means the walk is in state i with probability x . i n x1.  i i1 www.ThesisScientist.com 36Introduction to Information Retrieval Sec. 21.2.1 Change in probability vector  If the probability vector is x = (x , … x ) at 1 n this step, what is it at the next step?  Recall that row i of the transition prob. Matrix P tells us where we go next from state i.  So from x, our next state is distributed as xP 2 3  The one after that is xP , then xP , etc.  (Where) Does this converge? www.ThesisScientist.com 37Introduction to Information Retrieval Sec. 21.2.2 How do we compute this vector?  Let a = (a , … a ) denote the row vector of steady- 1 n state probabilities.  If our current position is described by a, then the next step is distributed as aP.  But a is the steady state, so a=aP.  Solving this matrix equation gives us a.  So a is the (left) eigenvector for P.  (Corresponds to the “principal” eigenvector of P with the largest eigenvalue.)  Transition probability matrices always have largest eigenvalue 1. www.ThesisScientist.com 38Introduction to Information Retrieval Link analysis: HITS www.ThesisScientist.com 39Introduction to Information Retrieval Sec. 21.3 Hyperlink-Induced Topic Search (HITS)  In response to a query, instead of an ordered list of pages each meeting the query, find two sets of inter- related pages:  Hub pages are good lists of links on a subject.  e.g., “Bob’s list of cancer-related links.”  Authority pages occur recurrently on good hubs for the subject.  Best suited for “broad topic” queries rather than for page-finding queries.  Gets at a broader slice of common opinion. www.ThesisScientist.com 40Introduction to Information Retrieval Sec. 21.3 Hubs and Authorities  Thus, a good hub page for a topic points to many authoritative pages for that topic.  A good authority page for a topic is pointed to by many good hubs for that topic.  Circular definition - will turn this into an iterative computation. www.ThesisScientist.com 41Introduction to Information Retrieval Sec. 21.3 The hope AT&T Alice Authorities Hubs ITIM Bob O2 Mobile telecom companies www.ThesisScientist.com 42Introduction to Information Retrieval Sec. 21.3 High-level scheme  Extract from the web a base set of pages that could be good hubs or authorities.  From these, identify a small set of top hub and authority pages; iterative algorithm. www.ThesisScientist.com 43Introduction to Information Retrieval Sec. 21.3 Base set  Given text query (say browser), use a text index to get all pages containing browser.  Call this the root set of pages.  Add in any page that either  points to a page in the root set, or  is pointed to by a page in the root set.  Call this the base set. www.ThesisScientist.com 44Introduction to Information Retrieval Sec. 21.3 Visualization Root set Base set Get in-links (and out-links) from a connectivity server www.ThesisScientist.com 45Introduction to Information Retrieval Sec. 21.3 Distilling hubs and authorities  Compute, for each page x in the base set, a hub score h(x) and an authority score a(x).  Initialize: for all x, h(x)1; a(x) 1;  Iteratively update all h(x), a(x); Key  After iterations  output pages with highest h() scores as top hubs  highest a() scores as top authorities. www.ThesisScientist.com 46Introduction to Information Retrieval Sec. 21.3 Iterative update  Repeat the following updates, for all x: h(x) a(y) x  xy a(x) h(y) x  yx www.ThesisScientist.com 47Introduction to Information Retrieval Sec. 21.3 Scaling  To prevent the h() and a() values from getting too big, can scale down after each iteration.  Scaling factor doesn’t really matter:  we only care about the relative values of the scores. www.ThesisScientist.com 48Introduction to Information Retrieval Sec. 21.3 How many iterations?  Claim: relative values of scores will converge after a few iterations:  in fact, suitably scaled, h() and a() scores settle into a steady state  proof of this comes later.  In practice, 5 iterations get you close to stability. www.ThesisScientist.com 49Introduction to Information Retrieval Sec. 21.3 Proof of convergence  nn adjacency matrix A:  each of the n pages in the base set has a row and column in the matrix.  Entry A = 1 if page i links to page j, else = 0. ij 1 2 3 1 2 1 0 1 0 2 1 1 1 3 1 0 0 3 www.ThesisScientist.com 50Introduction to Information Retrieval Sec. 21.3 Hub/authority vectors  View the hub scores h() and the authority scores a() as vectors with n components.  Recall the iterative updates h(x) a(y)  xy a(x) h(y)  yx www.ThesisScientist.com 51Introduction to Information Retrieval Sec. 21.3 Rewrite in matrix form t  h=Aa. Recall A is the t  a=A h. transpose of A. t t Substituting, h=AA h and a=A Aa. t Thus, h is an eigenvector of AA and a is an t eigenvector of A A. Further, our algorithm is a particular, known algorithm for computing eigenvectors: the power iteration method. Guaranteed to converge. www.ThesisScientist.com 52Introduction to Information Retrieval Sec. 21.3 Issues  Topic Drift  Off-topic pages can cause off-topic “authorities” to be returned  E.g., the neighborhood graph can be about a “super topic”  Mutually Reinforcing Affiliates  Affiliated pages/sites can boost each others’ scores  Linkage between affiliated pages is not a useful signal www.ThesisScientist.com 53Introduction to Information Retrieval Resources  IIR Chap 21  http://www2004.org/proceedings/docs/1p309.pdf  http://www2004.org/proceedings/docs/1p595.pdf  http://www2003.org/cdrom/papers/refereed/p270/ kamvar-270-xhtml/index.html  http://www2003.org/cdrom/papers/refereed/p641/ xhtml/p641-mccurley.html  The WebGraph framework I: Compression techniques (Boldi et al. 2004) www.ThesisScientist.com 54
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