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

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Where Do Matrices Come From?

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Computer Science Graphs: G = (V,E)

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Internet Graph

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View Internet Graph on Spheres

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Graphs in Scientific Computing

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Resource Allocation Graph

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Road Map

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Matrices Representation of graphs Adjacency matrix:

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Adjacency Matrix: 1 2 3 4 5

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

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1 2 3 4 5 Laplacian of Graphs

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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

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

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Random walks Transition Matrix 1 2 3 4 5 6

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

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Other Matrices Projections Rotations Permutations Reflections

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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

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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

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Term Occurrence Matrix

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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

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Matrix in Image Processing

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

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Random walks Transition Matrix 1 2 3 4 5 6

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