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Local Approximation of PageRank and Reverse PageRank

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Local Approximation of PageRank and Reverse PageRank

Li-Tal Mashiach

Advisor: Dr. Ziv Bar-Yossef

13/03/08

- Review of PageRank
- Local PageRank approximation
- Algorithm
- Lower bounds
- PageRank vs. Reverse PageRank
- Applications of Reverse PageRank

Most search engines analyze the hyperlink structure to order search results

PageRank

- Important measure of ranking for all major search engines

Base rank

Sum of the

in-neighbors’ ranks

Rank divided among

all out-neighbors

Damping factor

- A random surfer is visiting the web:
- With probability , selects a random out-link
- With probability jumps to a random web page

- Run power method
- Initialize:
- Repeat until convergence:

- Challenges:
- Holding the whole web graph
- Multiplying a matrix by a vector

Global PR calculates PR to all pages

Sometime we are interested in the PR of a small number of pages

- Person interested in the PR of his homepage
- Online business is interested in the PR of his own website and his competitors’ website
Do we need to calculate the PR of the whole graph for that?

- Given: local access to a directed graph G and target node
- Output: PR(u)
- local access:
- Cost: Number of queries to the link server

Link Server

Review of PageRank

Local PageRank approximation

Algorithm

Lower bounds

PageRank vs. Reverse PageRank

Applications of Reverse PageRank

- inft(v,u) – the fraction of the PR score of v that flows to u on paths of length t

v

u2

u1

u

- PRr(u) – PR score that flows into u from nodes at distance at most r from u
Theorem:

v

u2

u1

u

- Goal: calculate PRr(u) for a sufficiently large r
- Algorithm:
- Crawl backwards the sub-graph of radius r around u
- For each node v at layer t calculate the inft(v,u)
- Sum up the weighted influence values

v

w1

w2

u

u

Heuristic to improve the cost

Prune all nodes whose influence is below some threshold

Was shown empirically to be sometimes better [Chen, Gan, Suel, 2004]

u

- This algorithm requires at most queries
- r – number of iterations until the PR random walk almost converges
- d – maximum in-degree of the graph

- In case of slow PR convergence or high in-degree, the algorithm is not feasible

- In the web graph there are a lot of web pages with high in-degree
- Conclusion: The algorithm is frequently unsuitable for the web graph
- Is this a limitation of this
specific algorithm only?

- Local PR approx. is hard for graphs with:
- High in-degree nodes
- Slow convergence of the PR random walk

x1

x2

x3

xm

- By reduction from the OR problem

Input:

Output:

queries are needed even for

randomized algorithms

1

1

0

m

X=

Gx=

…

…

….

…

u

- A - Alg. that calculates local PR
- B - Alg. that computes the OR function

1

1

0

m

X=

Gx=

Claim 1: Let |x| be the

number of 1’s in x. Then,

…

…

….

…

u

Claim 2: When ,

- Given an input x, B simulates A on Gx, u
- If PRx(u) ≥ p1 => OR=1
- If PRx(u) ≤ p0 => OR=0
- It means that the maximum number of queries A uses ≥

- Local PageRank approximation is frequently infeasible on the web graph

- The local approximation algorithm should perform better on the Reverse Web Graph

280,000 page crawl of the www.stanford.edu domain

22,000 page crawl of the www.cnn.com site

In-deg: 38,606Out-deg: 255

Local RPR app.

Novel app.

TrustRank

Influencers in social networks

Hub web pages

Measuring semantic relatedness

Finding crawl seeds

Goal: Market a new product to be adopted by a large fraction of a social network

Method:

- Initially target a few influential members
- Trigger a word of mouth process
- Results in a large number of users
How should we choose these seed members?

- Nodes with high RPR
- Have short paths to many other nodes in the network
- Frequently the only gateways to these nodes

4-level BFS crawl

1-level BFS crawl

www.Livejournal.com, 3.5 million nodes

Goal: Find good starting points for search

- Difficult to formulate queries
- Broad search tasks
- Need to understand the surrounding context
Method: Find pages with short paths to many relevant pages

- High RPR pages tend to have short paths to many authorities

Fraction of hubs in the top 20 results for the queries:

1. “computer scientists”

2. “global warming”

3. “folk dancing”

4. “queen Elizabeth”

Meta-search engine over Yahoo! search

Goal: Find the relatedness between two concepts

- For Natural language processing applications
Method: Use a taxonomy like the ODP or Wikipedia

b is a strong sub-concept of a in a taxonomy if

- there are many short paths from a to b
RPR- measure of b as sub-concept of a

RPR Similarity- two concepts will be similar in case they have significant overlap between their RPR vectors

- similarity between the vectors RPRa and RPRb

Relatedness to “Einstein”

Relatedness to “Computer”

Agriculture

Physics Prize

Newton Isaac

Internet

0.6

0.6

-0.4

www.dmoz.org taxonomy

WordSimilarity-353

Goal: Discover quickly new content on the web while incurring as little overhead as possible

- Overhead: old pages / new pages
Method: Find good seeds

- A page p has high RPR if
- Many pages are reachable from p by short paths
- These pages are not reachable from many other pages

u

Known page

v

Unknown page

Fraction of new pages discovered

Overhead

WebBase project, two crawls of ~1,000,000 pages, one week apart

4-level BFS crawl

Two graph properties make local PageRank approximation hard

The Web Graph is not suitable for

local PR approximation

The Reverse Web graph is suitable

for local PR approximation

RPR finds nodes that

- have short paths to many other nodes
- frequently the only gateways to these nodes
Applications of RPR

x1

x2

x3

xm

- By reduction from the majority-by-a-margin problem

Input:

Output: the majority

At least queries are needed

1

1

0

m

X=

Gx=

W1

W2

Wm

V1

V2

V3

u

- A - Alg. that calculates local PR
- B - Alg. that computes majority-by-a-margin

1

1

0

m

X=

Claim 1: Let |x| be the

number of 1’s in x. Then,

Gx=

W1

W2

Wm

V1

V2

V3

u

Claim 2: When ,

- Given an input x, B simulates A on Gx, u
- If PRx(u) ≥ p1 => The majority bit of x is 1
- If PRx(u) ≤ p0 => The majority bit of x is 0
- It means that the maximum number of queries A uses ≥

x1

x2

x3

xm

- By reduction from the OR problem

Input:

Output:

queries are needed even for

randomized algorithms

0

1

0

m

X=

Gx=

Sm

S1

……

T

- A - Alg. that calculates local PR
- B - Alg. that computes the OR function

u

0

1

0

m

X=

Gx=

Claim 1: Let |x| be the

number of 1’s in x. Then,

Sm

S1

……

T

Claim 2: When ,

u

- Given an input x, B simulates A on Gx, u
- If PRx(u) ≥ p1 => OR=1
- If PRx(u) ≤ p0 => OR=0
- It means that the maximum number of queries A uses ≥

x1

x2

x3

xm

- By reduction from the majority-by-a-margin problem

Input:

Output: the majority

At least queries are needed

1

1

0

m

X=

w1

w2

w3

w4

Gx=

wm-1

wm

……

- A - Alg. that calculates local PR
- B - Alg. that computes majority-by-a-margin

……

……

……

u

1

1

0

m

X=

Claim 1: Let |x| be the

number of 1’s in x. Then,

w1

w2

w3

w4

wm-1

wm

……

……

……

……

Claim 2: When ,

u

- Given an input x, B simulates A on Gx, u
- If PRx(u) ≥ p1 => The majority bit of x is 1
- If PRx(u) ≤ p0 => The majority bit of x is 0
- It means that the maximum number of queries A uses ≥