Prestige Seeley, 1949 Brin

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Prestige(Seeley, 1949 Brin Page, 1997
Kleinberg,1997)

• Use edge-weighted, directed graphs to model
  social networks
• Status/Prestige
• In-degree is a good first-order indicator

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Notations

• Document citation graph,
• Node adjacency matrix E
• Ei,j 1 iff document i cites document j, and
  zero otherwise.
• Prestige pv associated with every node v
• Prestige vector over all nodes p

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Fixpoint Prestige Vector

• Confer to all nodes v the sum total of prestige
  of all u which links to v
• Gives a new prestige score p
• Fixpoint for prestige vector
• iterative assignment
• Fixpoint principal eigenvector of E
• Variants attenuation factor

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Centrality

• Graph-based notions of centrality
• Distance d(u,v) number of links between u and
  v0
• Radius of node u is
• Center of the graph is
• Example
• Influential papers in an area of research by
  looking for papers u with small r(u)
• No single measure is suited for all applications

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Co-citation

• v and w are said to be co-cited by u.
• If document u cites documents v and w
• Ei,j document citation matrix
• gt ETE co-citation index matrix
• Indicator of relatedness between v and w.
• Clustering
• Using above pair-wise relatedness measure in a
  clustering algorithm

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MDS Map of WWW Co-citationsSocial structure of
Web communities concerning Geophysics, climate,
remote sensing, and ecology. The cluster labels
are generated manually. Courtesy Larson
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The surfing model

• Correspondence between surfer model and the
  notion of prestige
• Page v has high prestige if the visit rate is
  high
• This happens if there are many neighbors u with
  high visit rates leading to v
• Deficiency
• Web graph is not strongly connected
• Only a fourth of the graph is !
• Web graph is not aperiodic
• Rank-sinks
• Pages without out-links
• Directed cyclic paths

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Surfing Model Simple fix

• Two way choice at each node
• With probability d (0.1 lt d lt 0.2), the surfer
  jumps to a random page on the Web.
• With probability 1d the surfer decides to
  choose, uniformly at random, an out-neighbor
• MODIFIED EQUATION 7.9
• Direct solution of eigen-system not feasible.

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Solution Power Iterations
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PageRank Architecture at Google

• Ranking of pages more important than exact values
  of pi
• Convergence of page ranks in 52 iterations for a
  crawl with 322 million links.
• Pre-compute and store the PageRank of each page.
• PageRank independent of any query or textual
  content.

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• Ranking scheme combines PageRank with textual
  match
• Unpublished
• Many empirical parameters, human effort and
  
• regression testing.
• Criticism Ad-hoc coupling and decoupling
  
• between relevance and prestige

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HITS Hyperlink Induced Topic Search

• Relies on query-time processing
• To select base set Vq of links for query q
  constructed by
• selecting a sub-graph R from the Web (root set)
  relevant to the query
• selecting any node u which neighbors any r \in R
  via an inbound or outbound edge (expanded set)
• To deduce hubs and authorities that exist in a
  sub-graph of the Web
• Every page u has two distinct measures of merit,
  its hub score hu and its authority score au.
• Recursive quantitative definitions of hub and
  authority scores

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Use text-based search engine to create a root set
of matching documents Expand root set to form
base set context graph of depth 1 additional
heuristics
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Query dependent input
Root Set
OUT
IN
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Query dependent input
Root Set
OUT
IN
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Query dependent input
Base Set
Root Set
OUT
IN
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• Associate two numerical scores with each document
  in a hyperlinked collection authority score and
  hub score
• Authorities most definitive information sources
  (on a specific topic)
• Like conference papers (new ideas)
• Hubs most useful compilation of links to
  authoritative documents
• Like journal papers or books (consolidate or
  
• survey significant research)

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J. Kleinberg. Authoritative sources in a
hyperlinked environment. Proc. 9th ACM-SIAM
Symposium on Discrete Algorithms, 1998

• Basic presumptions
• Creation of links indicates judgment conferred
  authority, endorsement
• Authority is not conferred directly from page to
  
• page, but rather mediated through hub nodes
• authorities may not be linked directly but
• through co-citation
• Example major car manufacturer pages will not
  point to each other, but there may be hub pages
  that compile links to such pages

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Hub Authority Scores

• Hubs and authorities exhibit what could be
  called a mutually reinforcing relationship a
  good hub is a page that points to many good
  authorities a good authority is a page that is
  pointed to by many good hubs Kleinberg 1999

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The HITS algorithm. h and aare L1 vector
norms
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Iterative Score Computation (1)

• Translate mutual relationship
• into iterative update equations

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Iterative Score Computation (2)

• Matrix notation

Adjacency matrix
Score vectors
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Iterative Score Computation (3)

• Condense into a single update equation (e.g.)
• Question of convergence (ignore absolute scale)
• Notice resemblance with eigenvector equations

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Example

• Simple example graph
• Hub authority matrices
• Authority and Hub weights

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HITS Topic Distillation Process

• Send query to a text-based IR system and obtain
  the root-set.
• Expand the root-set by radius one to obtain an
  expanded graph.
• Run power iterations on the hub and authority
  scores together.
• Report top-ranking authorities and hubs.

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HITS Applications

• Clever model http//www.almaden.ibm.com/cs/k53/cl
  ever.html
• Fine-grained ranking Soumen WWW10
• Query Sensitive retrieving Krishna Bharat
  SIGIR98

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PageRank vs. HITS

• PageRank advantage over HITS
• Query-time cost is low
• HITS computes an eigenvector for every query
• Less susceptible to localized link-spam
• HITS advantage over PageRank
• HITS ranking is sensitive to query
• HITS has notion of hubs and authorities
• Topic-sensitive PageRanking Haveliwala WWW11
• Attempt to make PageRanking query sensitive

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HITS Discussion

• Pros
• Derives topic-specific authority scores
• Returns list of hubs in addition to authorities
• Computational tractable (due to focused
  sub-graph)
• Cons
• Sensitive to Web spam (artificially increasing
  hub and authority weight)
• Query dependence requires expensive context graph
  building step
• Topic drift dominant topic in base set may not
  be the intended one

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Relation between HITS, PageRank and LSI

• HITS algorithm running SVD on the hyperlink
  relation (source,target)
• LSI algorithm running SVD on the relation
  (term,document).
• PageRank on root set R gives same ranking as the
  ranking of hubs as given by HITS

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