PageRank

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# PageRank - PowerPoint PPT Presentation

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

Brin, Page

description: C. Faloutsos, CMU

Problem definition:
• Given a directed graph
• which are the most ‘important’ nodes?

2

1

3

4

5

C. Faloutsos

• Imagine a particle randomly moving along the edges (*)
• compute its steady-state probabilities

(*) with occasional random jumps

C. Faloutsos

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

2

1

3

4

5

C. Faloutsos

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 algorithm
• A p = p

A p = p

2

1

3

=

4

5

C. Faloutsos

(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 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
• 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

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

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

C. Faloutsos