lecture 16 graphs and matrices in practice eigenvalue and eigenvector shang-hua teng
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Lecture 16Graphs and Matrices in Practice
Eigenvalue and Eigenvector
Shang-Hua Teng
Where Do Matrices Come From?
Computer Science
• Graphs: G = (V,E)
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