Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele...

Network Analysis

Graphs as Matrices

Michele Coscia & Luca RossiITU København

Background:

Background:

Looking at a problem from a different

perspective

Sergey Brin & Larry Page

(1973-)

55 Million!

PageRank

PageRank● p of visiting v…

PageRank● p of visiting v…● … = p of visiting a neighbor u of v *

p of going from u to v


p of going from u to v● for v in V…


p of going from u to v● for v in V…● 55 million times!

PageRankG = (V, E)

PageRankG = (V, E) A = 1 if uv in E

PageRank

for v in V...

G = (V, E) A = 1 if uv in E

PageRank

for v in V... R := AR


~

PageRank



~

Row-normalized A

PageRank



~

Row-normalized A

Eigenvectorof A

Random Walks

Basics

A B C D E FA

B

C

D

E

F

A B

Basics

A B C D E FA

B

C

D

E

F

Do youknow each

other?

A B

Basics

A B C D E FA

B

C

D

E

F

Do youknow each

other?

A B

Yes! Yes!

Basics

A B C D E FA

B

C

D

E

F

Do youknow each

other?

A B

Yes! Yes!

X

X

Basics

Basics

1 2 3 4 5 6 7 8 9

Basics

1 2 3 4 5 6 7 8 9123456789

Basics

1 2 3 4 5 6 7 8 9123456789

Adjacency Matrix

Properties of Adjacency Matrices


3


3

3

2

1

Stochastic Adjacency

A B C D E FA

B

C

D

E

F

AX X X


A B C D E FA

B

C

D

E

F

Pick oneof your friends

at random

AX X X


A B C D E FA

B

C

D

E

F

Pick oneof your friends

at random

A

Mmm…E!

X X X

Stochastic Adjacency● p of going from

v to u

Stochastic Adjacency● p of going from

v to u

● A / A.sum(axis = 1)

Stochastic Adjacency● p of going from v

to u

● A / A.sum(axis = 1)

● Useful eigen-properties

Walk Probabilities

Walk probabilities

Walk probabilities

p(1→2) in two steps?

Walk probabilities


(p(1→5) * p(5→2)) + (p(1→9) * p(9→2))

Walk probabilities


(p(1→5) * p(5→2)) + (p(1→9) * p(9→2))

p(u→v)?

Powers of Stochastic

A


A A2


1

A2 A30

Stationary Distribution

Stationary Distribution● A∞ = π


● πA = π


● πA = π

● ~ PageRank

Disconnected Mazes

Disconnected Mazes

?

Multiple Components

A

Multiple Components

A30

Google’s Solution

Google’s Solution

Beamme up,Scotty!

Google’s Solution

Google’s Solution

A

Google’s Solution

A+ε

Stationary Distribution vs Degree


Corr = 1


Corr = 1π = k / k.sum()

Stationary Distribution vs PageRank

Graph Spectrum

A Recap on Eigenvectors

v

(x, y)


v

w = Av

(x’, y’)

(x, y)


v

w = Av

(x’, y’)(λx, λy)

(x, y)


v

w = Av

(x’, y’)

w = Av = λv

(λx, λy)

(x, y)


v

w = Av

(x’, y’)

w = Av = λv

(λx, λy)

Eigenvector

(x, y)


v

w = Av

(x’, y’)

w = Av = λv

(λx, λy)

Eigenvector

Eigenvalue

(x, y)

Right & Left

Av = λv

Right & Left

Av = λv Right Eigenvector

Right & Left

Av = λv

vA = vλ

Right Eigenvector

Right & Left

Av = λv

vA = vλ

Right Eigenvector

Left Eigenvector

Eigenvalues

λ1 ≥ λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n

Eigenvalues

λ1 ≥ λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n

1

Eigenvalues & Components


λ1 ≥ λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n


λ1 ≥ λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n

If both equal to 1


λ1 ≥ λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n

If both equal to 1Graph has two components


λ1 ≥ λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n


λ1


λ1 ≥ λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n


λ1

λ2


λ1 > λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n

Eigenvectors

λ1 ≥ λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n

Eigenvectors

λ1 ≥ λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n

v1 v

2 v

3 … v

n-2 v

n-1 v

n

Eigenvectors

λ1 ≥ λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n

v1 v

2 v

3 … v

n-2 v

n-1 v

n

“Largest” Eigenvector

Eigenvectors

vi

a

b

c

d

e

f

g

h

i

l

m

n

o

Eigenvectors & Components

λ1 ≥ λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n


λ1

λ2


λ1 ≥ λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n


λ1

λ2

v1 v

2


λ1 ≥ λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n


λ1

λ2

v1 v

2

a

a

a

a

a

a

0

0

0

0

0

0

0

0

0

b

b

b


λ1 > λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n


λ1 > λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n

v1


λ1 > λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n

v1

λ1


λ1 > λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n

v1

a

a

a

a

a

a

a

a

a

λ1

Left Largest Eigenvector


vL1 = π


vL1 = π

πA = π


vL1 = π

πA = π

πA = π1


vL1 = π

πA = π

πA = π1

λ1

Left & Right

Left & RightvL

1

Left & RightvL

1

v1

a

a

a

a

a

a

a

a

a

Left & RightvL

1

vR1

v1

a

a

a

a

a

a

a

a

a

Left & RightvL

1

vR1 A

Left & RightvL

1

vR1 AT

Left & RightvL

1

vR1 AT

A / A.sum(axis = 0)

Left & Right

AT

A / A.sum(axis = 0)

Left & Right

AT

A / A.sum(axis = 0)

vL1

vR1

Graph Laplacian

Degree Matrix

Degree MatrixZeros everywhere


i-th diagonal element...


i-th diagonal element... … is the degree of node i


i-th diagonal element... … is the degree of node i

D

Laplacian Matrix

D A

Laplacian Matrix

D AL = -

Laplacian Matrix

L = D - A

Laplacian Matrix

L = D - A

0

Eigenvalues

λ1 ≥ λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n

1

A

L

Eigenvalues

λ1 ≥ λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n

1

A

L λ1 ≤ λ

2 ≤ λ

3 ≤ … ≤ λ

n-2 ≤ λ

n-1 ≤ λ

n

Eigenvalues

λ1 ≥ λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n

1

A

L λ1 ≤ λ

2 ≤ λ

3 ≤ … ≤ λ

n-2 ≤ λ

n-1 ≤ λ

n

0

Eigenvalues

λ1 ≥ λ

2 ≥ λ

3 ≥ … ≥ λ

n-2 ≥ λ

n-1 ≥ λ

n

1

A

L λ1 ≤ λ

2 ≤ λ

3 ≤ … ≤ λ

n-2 ≤ λ

n-1 ≤ λ

n

0

# of 0s = # of components

Eigenvectors

Eigenvectors

v1

Eigenvectors

v1

a

a

a

a

a

a

a

a

a

Eigenvectors

v1

a

a

a

a

a

a

a

a

av

2

Eigenvectors

v1

a

a

a

a

a

a

a

a

av

2

.37

.37

.28

.28

.12

-.23

-.38

-.38

-.43

Eigenvectors

Eigenvectors

+

Eigenvectors

+-

Eigenvectors

Eigenvectors

2nd Eigenvector

Eigenvectors

2nd Eigenvector

3rd Eigenvector

Eigenvectors

Textbook Chapters● Adjacency Matrix

– Section 2.4● Stochastic, Graph Spectrum &

Laplacian– http://www.michelecoscia.com/?

page_id=1488● Done with Chapter 2!

– Section 2.11

Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele...

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