Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.
-
date post
19-Dec-2015 -
Category
Documents
-
view
219 -
download
0
Transcript of Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.
![Page 1: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/1.jpg)
Lecture 16Graphs and Matrices in Practice
Eigenvalue and Eigenvector
Shang-Hua Teng
![Page 2: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/2.jpg)
Where Do Matrices Come From?
![Page 3: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/3.jpg)
Computer Science
• Graphs: G = (V,E)
![Page 6: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/6.jpg)
View Internet Graph on Spheres
![Page 7: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/7.jpg)
Graphs in Scientific Computing
![Page 8: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/8.jpg)
Resource Allocation Graph
![Page 9: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/9.jpg)
Road Map
![Page 10: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/10.jpg)
Matrices Representation of graphs
Adjacency matrix: ( ) , # edgesij ijA a a ij
![Page 11: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/11.jpg)
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
![Page 12: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/12.jpg)
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
![Page 13: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/13.jpg)
21001
12100
01311
00121
10113
L
1
2
34
5
Laplacian of Graphs
![Page 14: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/14.jpg)
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
![Page 15: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/15.jpg)
Random walks
How long does it take to get completely lost?
![Page 16: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/16.jpg)
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
![Page 17: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/17.jpg)
Markov Matrix
• Every entry is non-negative
• Every column adds to 1
• A Markov matrix defines a Markov chain
![Page 18: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/18.jpg)
Other Matrices
• Projections
• Rotations
• Permutations
• Reflections
![Page 19: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/19.jpg)
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
![Page 20: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/20.jpg)
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
![Page 21: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/21.jpg)
Term Occurrence Matrix
![Page 22: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/22.jpg)
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
![Page 23: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/23.jpg)
Matrix in Image Processing
![Page 24: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/24.jpg)
Random walks
How long does it take to get completely lost?
0
0
0
0
0
1
![Page 25: Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d3b5503460f94a1599b/html5/thumbnails/25.jpg)
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