lecture 16 graphs and matrices in practice eigenvalue and eigenvector shang-hua teng

Lecture 16Graphs and Matrices in Practice

Eigenvalue and Eigenvector

Shang-Hua Teng

Where Do Matrices Come From?

Computer Science

• Graphs: G = (V,E)

Internet Graph

View Internet Graph on Spheres

Graphs in Scientific Computing

Resource Allocation Graph

Road Map

Matrices Representation of graphs

Adjacency matrix: ( ) , # edgesij ijA a a ij

Adjacency Matrix:

01001

10100

01011

00101

10110

A

1

2

34

5

edgean not is j)(i, if 0

edgean is j)(i, if 1ijA

Matrix of GraphsAdjacency Matrix:• If A(i, j) = 1: edge exists

Else A(i, j) = 0.

0101

0000

1100

00101 2

34

1

-3

3

2 4

21001

12100

01311

00121

10113

L

1

2

34

5

Laplacian of Graphs

Matrix of Weighted GraphsWeighted Matrix:• If A(i, j) = w(i,j): edge exists

Else A(i, j) = infty.

032

0

430

101 2

34

1

-3

3

2 4

Random walks

How long does it take to get completely lost?

Random walks Transition Matrix

1

2

345

6

02

1

4

100

2

13

10

4

1000

3

1

2

10

2

1

3

10

004

10

3

10

004

1

2

10

2

13

1000

3

10

P

Markov Matrix

• Every entry is non-negative

• Every column adds to 1

• A Markov matrix defines a Markov chain

Other Matrices

• Projections

• Rotations

• Permutations

• Reflections

Term-Document Matrix• Index each document (by human or by

computer)– fij counts, frequencies, weights, etc

m term

2 term

1 term

n docdoc21 doc

21

22221

1 1211

mnmm

n

n

fff

fff

fff

• Each document can be regarded as a point in m dimensions

Document-Term Matrix• Index each document (by human or by

computer)– fij counts, frequencies, weights, etc

m doc

2 doc

1 doc

n term2 term1 term

21

22221

1 1211

mnmm

n

n

fff

fff

fff

• Each document can be regarded as a point in n dimensions

Term Occurrence Matrix

c1 c2 c3 c4 c5 m1 m2 m3 m4 human 1 0 0 1 0 0 0 0 0 interface 1 0 1 0 0 0 0 0 0 computer 1 1 0 0 0 0 0 0 0 user 0 1 1 0 1 0 0 0 0 system 0 1 1 2 0 0 0 0 0 response 0 1 0 0 1 0 0 0 0 time 0 1 0 0 1 0 0 0 0 EPS 0 0 1 1 0 0 0 0 0 survey 0 1 0 0 0 0 0 0 1 trees 0 0 0 0 0 1 1 1 0 graph 0 0 0 0 0 0 1 1 1 minors 0 0 0 0 0 0 0 1 1

Matrix in Image Processing

Random walks

How long does it take to get completely lost?

0

0

0

0

0

1

Random walks Transition Matrix

1

2

345

6

0

0

0

0

0

1

02

1

4

100

2

13

10

4

1000

3

1

2

10

2

1

3

10

004

10

3

10

004

1

2

10

2

13

1000

3

10

100

P