1. Lecture 16 Graphs and Matrices in Practice Eigenvalue and Eigenvector Shang-Hua Teng

2. Where Do Matrices Come From?

3. Computer Science Graphs: G = (V,E)

4. Internet Graph

6. View Internet Graph on Spheres

7. Graphs in Scientific Computing

8. Resource Allocation Graph

9. Road Map

10. Matrices Representation of graphs Adjacency matrix:

11. Adjacency Matrix: 1 2 3 4 5

12. Matrix of Graphs Adjacency Matrix: If A(i, j) = 1: edge exists Else A(i, j) = 0. 12 34 1 -3 3 2 4

13. 1 2 3 4 5 Laplacian of Graphs

14. Matrix of Weighted Graphs Weighted Matrix: If A(i, j) = w(i,j): edge exists Else A(i, j) = infty. 12 34 1 -3 3 2 4

15. Random walks How long does it take to get completely lost?

16. Random walks Transition Matrix 1 2 3 4 5 6

17. Markov Matrix Every entry is non-negative Every column adds to 1 A Markov matrix defines a Markov chain

18. Other Matrices Projections Rotations Permutations Reflections

19. Term-Document Matrix Index each document (by human or by computer) –f ij counts, frequencies, weights, etc Each document can be regarded as a point in m dimensions

20. Document-Term Matrix Index each document (by human or by computer) –f ij counts, frequencies, weights, etc Each document can be regarded as a point in n dimensions

21. Term Occurrence Matrix

22. 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

23. Matrix in Image Processing

24. Random walks How long does it take to get completely lost?

25. Random walks Transition Matrix 1 2 3 4 5 6