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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

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

• 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.

C. Faloutsos