Recitation 3
Post on 01-Jan-2016
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DESCRIPTION
Transcript
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
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:
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
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
Randomness
• Stock Prices
• Games (Poker, Casino, etc)
• Biology: Evolution, Mutation
• Physics: Quantum Mechanics
• …
• Is the world deterministic or stochastic?
Review
• LDSDDS
• Uncover the secret: Google’s PageRank
• DeterministicStochastic
• That’s more fascinating
• Welcome to the Stochastic World!
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