Vincent Blondel and Paul Van Dooren CESAME, Universite Catholique de Louvain

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Web searching and graph similarityVincent
Blondel and Paul Van DoorenCESAME, Universite
Catholique de Louvainhttp//www.csam.ucl.ac.be/

• in collaboration with A. Gajardo, M. Heymans, P.
  Sennelart
• To appear in Siam Review
  
  Bari 6,September 2004
  

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The web graph

• Nodes web pages, Edges hyperlinks between
  pages
• 3 billion (Google searched 3,083,324,625 webpages
  in 2002)
• Average of 7 outgoing links

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The web graph

• Nodes web pages, Edges hyperlinks between
  pages
• 3 billion (Google searched 3,083,324,625 webpages
  in 2002)
• Average of 7 outgoing links
• Growth of a few
• every month

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Outline

• 1. Structure of the web
• 2. Methods for searching the web
• (Google PageRank and Kleinberg Hits)
• 3. Similarity in graphs
• 4. Application to synonym extraction

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Structure of the web

• Experiments two crawls in 1999 found a giant
  strongly connected
  component (core)
• Contains most prominent sites
• It contains 30 of all pages
• Average distance between nodes is 16
• Small world
• Ref Broder et al., Graph structure in the web,
  WWW9, 2000

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The web is a bowtie

• Ref The web is a bowtie, Nature, May 11, 2000

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In- and out-degree distributions

• Power law distribution number of pages of
  in-degree n is
• proportional to 1/n2.1 (Zipf law)

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A score for every page

• The score of a page is high if the page has many
  incoming
• links coming from pages with high page score
• One browses from page to page by following
  outgoing links
• with equal probability. Score frequency a page
  is visited.

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A score for every page

• The score of a page is high if the page has many
  incoming
• links coming from pages with high page score
• One browses from page to page by following
  outgoing links
• with equal probability. Score frequency a page
  is visited.
• some pages may have no outgoing links
• many pages have zero in-degree

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PageRank teleporting random score

• The surfer follows a path by choosing an outgoing
  link with probability
• p/dout(i) or teleports to a random web page with
  probability 0lt1-p lt1.
• Put the transition probability of i to j in a
  matrix M (bij1 if i?j)
• mij p bij /dout(i) (1-p)/n
• then the vector x of probability distribution on
  the nodes of the graph
• is the steady state vector of the iteration
  xk1MTxk i.e. the dominant
• eigenvector of the matrix MT (unique because of
  Perron-Frobenius)
• PageRank of node i is the (relative) size of
  element i of this vector

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Matlab News and Notes, October 2002
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and my own page rank ?

• use Google toolbar
• some top pages
• PageRank In-degree
• 1 http//www.yahoo.com 10 654,000
• 2 http//www.adobe.com 10 646,000
• 5 http//www.google.com 10 252,000
• 8 http//www.microsoft.com 10 129,000
• 12 http//www.nasa.gov 10 93,900
• 20 http//mit.edu 10 47,600
• 23 http//www.nsf.gov 10 39,400
• 26 http//www.inria.fr 10 17,400
• 72 http//www.stanford.edu 9 36,300

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Kleinbergs structure graph

• The score of a page is high if the page has
• many incoming links
• The score is high if the incoming links are
• from pages that have high scores

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Kleinbergs structure graph

• The score of a page is high if the page has
• many incoming links
• The score is high if the incoming links are
• from pages that have high scores
• This inspired Kleinbergs structure graph
• hub authority

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Good authorities for University Belgium
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A good hub for University Belgium
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Hub and authority scores

• Web pages have a hub score hj and an authority
  score aj which are
• mutually reinforcing
• pages with large hj point to pages with high aj
• pages with large aj are pointed to by pages with
  high hj
• hj ?
  S i(j?i) ai
• aj ? S i(i?j) hi
• or, using the adjacency matrix B of the graph
  (bji1 if j?i is an edge)
• h 0 B
  h h 1
• a k1 BT 0 a
  k a 0 1
• Use limiting vector a (dominant eigenvector of
  BTB) to rank pages

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Extension to another structure graph

• Give three scores to each web page begin b,
  center c, end e
• b
  c e
• Use again mutual reinforcement to define the
  iteration
• bj ?
  S i(j?i) ci
• cj ? S i(i?j) bi
  S i(j?i) ei
• ej ?
  S i(i?j) ci
• Defines a limiting vector for the iteration
• 
  b 0 B
  0
• xk1 M xk, x0 1 where
  x c , M BT 0 B
• 
  e 0 BT
  0

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Towards arbitrary graphs

• For the graph ? A
  and M
• For the graph ? ? A
  and M
• Formula for M for two arbitrary graphs GA and GB
  
• M A B
  AT BT
• With xk vec(Xk) iteration xk1 M xk is
  equivalent to Xk1 BXk ATBT Xk A

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Convergence ?

• The (normalized) sequence
• Zk1 (BZk ATBT Zk A)/ BZk ATBT Zk AF
• has two fixed points Zeven and Zodd for every
  Z0gt0
• Similarity matrix S lim k?8 Z2k , Z0 1
• Si,j is the similarity score between Vj (A) and
  Vi (B)
• Properties (generic)
• ?S BSAT BTSA, ?BSATBTSAF
• Fixed point of largest 1-norm
• Robust fixed point for ?eSe BSeAT BTSeA
  e1Se1
• Linear convergence (power method for sparse M)

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Bow tie example

• h a h a
• S S
• if mgtn if ngtm
• not satisfactory

graph A h ? a
graph B 2
1 n1
nm1
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Bow tie example

• b c e
• S
• central score is good

graph A b ? ? e c
graph B 2
1 n1
nm1
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Other properties

• Central score is a dominant eigenvector of
  BBTBTB
• (cfr. hub score of BBT and authority score of
  BTB)
• Similarity matrix of a graph with itself is
  square and semi-definite.
• Path graph ? ?
  Cycle graph

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The dictionary graph

• OPTED, based on Websters unabridged dictionary
• http//msowww.anu.edu.au/ralph/OPTED
• Nodes words present in the dictionary 112,169
  nodes
• Edge (u,v) if v appears in the definition of u
  1,398,424 edges
• Average of 12 edges per node

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In and out degree distribution

• Very similar to web (power law)
• Words with highest in degree
• of, a, the, or, to, in
• Words with null out degree
• 14159, Fe3O4, Aaron,
• and some undefined or misspelled words

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Neighborhood graph

• is the subset of vertices used for finding
  synonyms
• it contains all parents and children of the node
• neighborhood graph
  of likely
• Central uses this sub-graph to rank
  automatically synonyms
• Comparison with Vectors, ArcRank (automatic)
• Wordnet, Microsoft
  Word (manual)

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Disappear
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Science
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Sugar
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Conclusion

• New notion of similarity between vertices of a
  graph
• Easy to compute start from X0 1 and take even
  normalized
• iterates of Xk1BXkATBTXkA
• Potential use for data-mining, classification,
  clustering
• Successful implementation for the French
  dictionary Le petit Robert
• Applications in texts, internet, reference lists,
  telephone networks,
• bipartite graphs (Melnik, Widom, )
• Different from sub-graph problems !

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Distribution of calls received
Number of customers
Number of calls received
Example 2000 people have received 100 calls