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PageRank. Brin, Page description: C. Faloutsos, CMU. Problem definition:. Given a directed graph which are the most ‘important’ nodes?. 2. 1. 3. 4. 5. google/Page-rank algorithm. Imagine a particle randomly moving along the edges (*) compute its steady-state probabilities

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PageRank

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Pagerank l.jpg

PageRank

Brin, Page

description: C. Faloutsos, CMU


Problem definition l.jpg

Problem definition:

  • Given a directed graph

  • which are the most ‘important’ nodes?

2

1

3

4

5

C. Faloutsos


Google page rank algorithm l.jpg

google/Page-rank algorithm

  • Imagine a particle randomly moving along the edges (*)

  • compute its steady-state probabilities

    (*) with occasional random jumps

C. Faloutsos


Google page rank algorithm4 l.jpg

google/Page-rank algorithm

  • that is: given a Markov Chain, compute the steady state probabilities p1 ... p5

2

1

3

4

5

C. Faloutsos


Simplified pagerank algorithm l.jpg

2

1

3

4

5

(Simplified) PageRank algorithm

  • Let W be the transition matrix (= adjacency matrix); let A be WT, and column-normalized - then

From

A

To

=

C. Faloutsos


Simplified pagerank algorithm6 l.jpg

(Simplified) PageRank algorithm

  • A p = p

A p = p

2

1

3

=

4

5

C. Faloutsos


Simplified pagerank algorithm7 l.jpg

(Simplified) PageRank algorithm

  • A p = 1 * p

  • thus, p is the eigenvector that corresponds to the highest eigenvalue(=1, since the matrix is column-normalized)

C. Faloutsos


Simplified pagerank algorithm8 l.jpg

(Simplified) PageRank algorithm

  • In short: imagine a particle randomly moving along the edges

  • compute its steady-state probabilities

    Full version of algo: with occasional random jumps

C. Faloutsos


Full algorithm l.jpg

Full Algorithm

  • With probability 1-c, fly-out to a random node

  • Then, we have

    p = c Ap + (1-c)/n 1 =>

    p = (1-c)/n [I - c A] -1 1

C. Faloutsos


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Impact - current research

  • multi-billion $ company

  • over 2,500 citations (Google scholar)

  • Topic-Sensitive PageRank [Haveliwala+]

  • TrustRank [Gyongyi+]

  • Efficient computation

  • ObjectRank [Papakonstantinou+]

  • centerPiece subgraphs [Tong+]

  • ...

C. Faloutsos


References l.jpg

Brin, S. and L. Page (1998). Anatomy of a Large-Scale Hypertextual Web Search Engine. 7th Intl World Wide Web Conf.

References

C. Faloutsos


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