Recitation 3
Steve Gu
Jan 31 2008

Outline
• Part I: Review of LDSDDS
– Linear, Deterministic, Stationary, Discrete,
Dynamic System
– Example: Google’s PageRank
• Part II: From Deterministic to Stochastic
– Randomness
– Some histograms

Part I

Review of LDSDDS
X(0) = x
0 y(n) = H  X (n)
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Review of LDSDDS
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3
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 y y
1
2
3
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1
( ) y y
2
3
For example: x (n +1)
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1,1
2,1
3,1 f f f
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3,2 f f f
1,3
2,3
3,3
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2 x (n)
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 g g
1,1
2,1
3,1 g g g
1,2
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3,2 g g g
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2,3
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1 x (n)
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1 u (n)
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Review of LDSDDS
• Interested?
• Confused?
• Doubted?
• Bored?
• Hey! Let’s take a real example

PageRank
• PageRank was developed at Stanford
University by Larry Page (hence the name
Page -Rank [1] ) and later Sergey Brin as part of a research project about a new kind of search engine. The project started in
1995 and led to a functional prototype, named Google, in 1998

How to rank the importance of web pages?
PageRank

PageRank http://en.wikipedia.org/wiki/Image:PageRanks-Example.svg

PageRank: Modelling Votes
PR(u) =  v link to u
PR(v)
L(v)
 PR(v) is the
PageRank of v
 L(v) is the number of pages linked to v
 PR(u) is a collection of votes by pages linked to it!

• For example:
B
PageRank
A receives 3 votes
B receives 1 votes
C receives 1 votes
D receives none
A
D
C
PR(A) = PR(B) +
PR(B) =
PR(C) =
PR(C)
2
PR(D)
2
PR(D) = 0
2 2

PageRank: Dynamic Systems?
For N pages, say p1,…,pN
Write the Equation to compute PageRank as: where l(i,j) is define to be:

X =
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PageRank: Dynamic Systems?
• Written in Matrix Form:
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1
PR(p ,n +1)
N-1
PR(p ,n +1)
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 l(1,1) l(1,2) l(2,1) l(2,2) l(N,1) l(1,N) l(2,N) l(N,N - 1) l(N,N)
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F Look familiar?
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PageRank: Dynamic Systems?
• Usually there is a damping factor d, which is used to guarantee convergence, that is:
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1
PR(p ,n +1)
N-1
PR(p ,n +1)
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=
(1- d)
N
1
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1
1
+ d
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 l(1,1) l(1,2) l(2,1) l(2,2) l(N,1) l(1,N) l(2,N) l(N,N - 1) l(N,N)
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)
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PageRank: Dynamic Systems!
• PageRank is fully described by a LDSDDS
• There is no magic here!
• Ideas change the world (e.g. Google)
• LDSDDS is simple
• LDSDDS is powerful
• LDSDDS is useful
• LDSDDS is beautiful

Part II
From Deterministic to Stochastic

Randomness

Randomness
• Stock Prices
• Games (Poker, Casino, etc)
• Biology: Evolution, Mutation
• Physics: Quantum Mechanics
• …
• Is the world deterministic or stochastic?

Some Common Histograms

Review
• LDSDDS
• Uncover the secret: Google’s PageRank
• Deterministic  Stochastic
• That’s more fascinating
• Welcome to the Stochastic World!

• Thank you
• Q&A
The End