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

0X(0)=x

X(n+1)=F X(n)+G u(n)

y(n)=H (n)

Review of LDSDDS

1 1 1

2 2 2

3 3 3

1,1 1,2 1,3 1,1 1,2 1,31 1

2 2,1 2,2 2,3 2 2,1 2,2 2,3

3 33,1 3,2 3,3 3,1 3,2 3,3

x u y

X = x ,U = u ,Y = y

x u y

ff f g g gx (n+1) x (n)

x (n+1) = ff f x (n) + g g g

x (n+1) x (n)ff f g g g

( )

( )

( )

n

n

n

1

2

3

1,1 1,2 1,31 1

2 2,1 2,2 2,3 2

3 33,1 3,2 3,3

u (n)

u (n)

u (n)

h h hy x (n)

y = h h h x (n)

y x (n)h h h

For example:

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

PageRankHow to rank the importance of web pages?

PageRank

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

PageRank: Modelling Votes

v link to u

PR(v)PR(u)=

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!

PageRank

• For example:

A

B

D

C

A receives 3 votes

B receives 1 votes

C receives 1 votes

D receives none

PR(C) PR(D)PR(A)=PR(B)+ +

2 2PR(C)

PR(B)=2

PR(D)PR(C)=

2PR(D)=0

PageRank: Dynamic Systems?For N pages, say p1,…,pN

Write the Equation to compute PageRank as:

where l(i,j) is define to be:

PageRank: Dynamic Systems?

• Written in Matrix Form:

1

2

N-1

N

PR(p )

PR(p )

X =

PR(p )

PR(p )

1 1

2 2

N-1 N-1

N N

PR(p ,n+1) PR(p ,n)l(1,1) l(1,2) l(1,N)

PR(p ,n+1) PR(p ,n)l(2,1) l(2,2) l(2,N)

PR(p ,n+1) PR(p ,n)l(N,1) l(N,N-1) l(N,N)

PR(p ,n+1) PR(p ,n)

F

X(n+1)=F X(n)

Look familiar?

PageRank: Dynamic Systems?

• Usually there is a damping factor d, which is used to guarantee convergence, that is:

1 1

2 2

N-1 N-1

N N

PR(p ,n+1) PR(p ,n)1l(1,1) l(1,2) l(1,N)

PR(p ,n+1) PR(p ,n)1l(2,1) l(2,2) l(2,N)(1- d)

= +dN

PR(p ,n+1) PR(p ,n)1l(N,1) l(N,N-1) l(N,N)

PR(p ,n+1) PR(p ,n1

)

X(n+1)=F X(n)+G u(n)

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

• DeterministicStochastic

• That’s more fascinating

• Welcome to the Stochastic World!

The End

• Thank you

• Q&A