A1: Simple 4 node loop Verify that the crowd walks around in a bunch. Press RETURN... pagerank_test(): The digraph adjacency matrix A. [[0 1 0 0] [0 0 1 0] [0 0 0 1] [1 0 0 0]] Press RETURN Plot the digraph A. Press RETURN Define transition matrix T: [[0. 0. 0. 1.] [1. 0. 0. 0.] [0. 1. 0. 0.] [0. 0. 1. 0.]] Press RETURN Watch 100 people move around digraph, x <- T * x: 0 [100. 0. 0. 0.] 1 [ 0. 100. 0. 0.] 2 [ 0. 0. 100. 0.] 3 [ 0. 0. 0. 100.] 4 [100. 0. 0. 0.] 5 [ 0. 100. 0. 0.] 6 [ 0. 0. 100. 0.] 7 [ 0. 0. 0. 100.] 8 [100. 0. 0. 0.] 9 [ 0. 100. 0. 0.] Press RETURN Eigenvalues of T? [-1.+0.j 0.+1.j 0.-1.j 1.+0.j] Press RETURN A2: 4 node loop with 1 branch The initial crowd spreads out. pagerank_test(): The digraph adjacency matrix A. [[0 1 1 0] [0 0 1 0] [0 0 0 1] [1 0 0 0]] Press RETURN Plot the digraph A. Press RETURN Define transition matrix T: [[0. 0. 0. 1. ] [0.5 0. 0. 0. ] [0.5 1. 0. 0. ] [0. 0. 1. 0. ]] Press RETURN Watch 100 people move around digraph, x <- T * x: 0 [100. 0. 0. 0.] 1 [ 0. 50. 50. 0.] 2 [ 0. 0. 50. 50.] 3 [50. 0. 0. 50.] 4 [50. 25. 25. 0.] 5 [ 0. 25. 50. 25.] 6 [25. 0. 25. 50.] 7 [50. 12.5 12.5 25. ] 8 [25. 25. 37.5 12.5] 9 [12.5 12.5 37.5 37.5] 10 [37.5 6.25 18.75 37.5 ] 11 [37.5 18.75 25. 18.75] 12 [18.75 18.75 37.5 25. ] 13 [25. 9.38 28.12 37.5 ] 14 [37.5 12.5 21.88 28.12] 15 [28.12 18.75 31.25 21.88] 16 [21.88 14.06 32.81 31.25] 17 [31.25 10.94 25. 32.81] 18 [32.81 15.62 26.56 25. ] 19 [25. 16.41 32.03 26.56] 20 [26.56 12.5 28.91 32.03] 21 [32.03 13.28 25.78 28.91] 22 [28.91 16.02 29.3 25.78] 23 [25.78 14.45 30.47 29.3 ] 24 [29.3 12.89 27.34 30.47] 25 [30.47 14.65 27.54 27.34] 26 [27.34 15.23 29.88 27.54] 27 [27.54 13.67 28.91 29.88] 28 [29.88 13.77 27.44 28.91] 29 [28.91 14.94 28.71 27.44] Press RETURN Eigenvalues of T? [ 1. +0.j -0.18+0.86j -0.18-0.86j -0.65+0.j ] Press RETURN A3: 7 nodes, with a source, a sink, and an island option 0: "sinks" drain. People who go to room F disappear. pagerank_test(): The digraph adjacency matrix A. [[0 1 0 0 0 1 0] [0 0 1 0 0 0 0] [1 0 0 1 0 0 0] [1 0 0 0 0 0 0] [0 0 1 0 0 0 0] [0 0 0 0 0 0 0] [0 0 0 0 0 0 0]] Press RETURN Plot the digraph A. Press RETURN Define transition matrix T: [[0. 0. 0.5 1. 0. 0. 0. ] [0.5 0. 0. 0. 0. 0. 0. ] [0. 1. 0. 0. 1. 0. 0. ] [0. 0. 0.5 0. 0. 0. 0. ] [0. 0. 0. 0. 0. 0. 0. ] [0.5 0. 0. 0. 0. 0. 0. ] [0. 0. 0. 0. 0. 0. 0. ]] Press RETURN Watch 100 people move around digraph, x <- T * x: 0 [100. 0. 0. 0. 0. 0. 0.] 1 [ 0. 50. 0. 0. 0. 50. 0.] 2 [ 0. 0. 50. 0. 0. 0. 0.] 3 [25. 0. 0. 25. 0. 0. 0.] 4 [25. 12.5 0. 0. 0. 12.5 0. ] 5 [ 0. 12.5 12.5 0. 0. 12.5 0. ] 6 [ 6.25 0. 12.5 6.25 0. 0. 0. ] 7 [12.5 3.12 0. 6.25 0. 3.12 0. ] 8 [6.25 6.25 3.12 0. 0. 6.25 0. ] 9 [1.56 3.12 6.25 1.56 0. 3.12 0. ] 10 [4.69 0.78 3.12 3.12 0. 0.78 0. ] 11 [4.69 2.34 0.78 1.56 0. 2.34 0. ] 12 [1.95 2.34 2.34 0.39 0. 2.34 0. ] 13 [1.56 0.98 2.34 1.17 0. 0.98 0. ] 14 [2.34 0.78 0.98 1.17 0. 0.78 0. ] Press RETURN Eigenvalues of T? [ 0. +0.j 0.82+0.j -0.12+0.72j -0.12-0.72j -0.57+0.j 0. +0.j 0. +0.j ] Press RETURN A3: 7 nodes, with a source, a sink, and an island option 1: "sinks" stagnate. People who go to room F can never leave. pagerank_test(): The digraph adjacency matrix A. [[0 1 0 0 0 1 0] [0 0 1 0 0 0 0] [1 0 0 1 0 0 0] [1 0 0 0 0 0 0] [0 0 1 0 0 0 0] [0 0 0 0 0 0 0] [0 0 0 0 0 0 0]] Press RETURN Plot the digraph A. Press RETURN Define transition matrix T: [[0. 0. 0.5 1. 0. 0. 0. ] [0.5 0. 0. 0. 0. 0. 0. ] [0. 1. 0. 0. 1. 0. 0. ] [0. 0. 0.5 0. 0. 0. 0. ] [0. 0. 0. 0. 0. 0. 0. ] [0.5 0. 0. 0. 0. 1. 0. ] [0. 0. 0. 0. 0. 0. 1. ]] Press RETURN Watch 100 people move around digraph, x <- T * x: 0 [100. 0. 0. 0. 0. 0. 0.] 1 [ 0. 50. 0. 0. 0. 50. 0.] 2 [ 0. 0. 50. 0. 0. 50. 0.] 3 [25. 0. 0. 25. 0. 50. 0.] 4 [25. 12.5 0. 0. 0. 62.5 0. ] 5 [ 0. 12.5 12.5 0. 0. 75. 0. ] 6 [ 6.25 0. 12.5 6.25 0. 75. 0. ] 7 [12.5 3.12 0. 6.25 0. 78.12 0. ] 8 [ 6.25 6.25 3.12 0. 0. 84.38 0. ] 9 [ 1.56 3.12 6.25 1.56 0. 87.5 0. ] 10 [ 4.69 0.78 3.12 3.12 0. 88.28 0. ] 11 [ 4.69 2.34 0.78 1.56 0. 90.62 0. ] 12 [ 1.95 2.34 2.34 0.39 0. 92.97 0. ] 13 [ 1.56 0.98 2.34 1.17 0. 93.95 0. ] 14 [ 2.34 0.78 0.98 1.17 0. 94.73 0. ] Press RETURN Eigenvalues of T? [ 1. +0.j 0.82+0.j -0.12+0.72j -0.12-0.72j -0.57+0.j 0. +0.j 1. +0.j ] Press RETURN A3: 7 nodes, with a source, a sink, and an island option 2: "sinks" evaporate. People who go to room F next visit any random room. pagerank_test(): The digraph adjacency matrix A. [[0 1 0 0 0 1 0] [0 0 1 0 0 0 0] [1 0 0 1 0 0 0] [1 0 0 0 0 0 0] [0 0 1 0 0 0 0] [0 0 0 0 0 0 0] [0 0 0 0 0 0 0]] Press RETURN Plot the digraph A. Press RETURN Define transition matrix T: [[0. 0. 0.5 1. 0. 0.14 0.14] [0.5 0. 0. 0. 0. 0.14 0.14] [0. 1. 0. 0. 1. 0.14 0.14] [0. 0. 0.5 0. 0. 0.14 0.14] [0. 0. 0. 0. 0. 0.14 0.14] [0.5 0. 0. 0. 0. 0.14 0.14] [0. 0. 0. 0. 0. 0.14 0.14]] Press RETURN Watch 100 people move around digraph, x <- T * x: 0 [100. 0. 0. 0. 0. 0. 0.] 1 [ 0. 50. 0. 0. 0. 50. 0.] 2 [ 7.14 7.14 57.14 7.14 7.14 7.14 7.14] 3 [37.76 5.61 16.33 30.61 2.04 5.61 2.04] 4 [39.87 19.97 8.75 9.26 1.09 19.97 1.09] 5 [16.64 22.94 24.07 7.38 3.01 22.94 3.01] 6 [23.13 12.03 29.66 15.74 3.71 12.03 3.71] 7 [32.82 13.81 17.98 17.08 2.25 13.81 2.25] 8 [28.36 18.71 18.35 11.29 2.29 18.71 2.29] 9 [23.46 17.18 24. 12.18 3. 17.18 3. ] 10 [27.06 14.61 23.06 14.88 2.88 14.61 2.88] 11 [28.91 16.03 20. 14.03 2.5 16.03 2.5 ] 12 [26.68 17.1 21.18 12.65 2.65 17.1 2.65] 13 [26.05 16.16 22.57 13.41 2.82 16.16 2.82] 14 [27.41 15.74 21.69 14. 2.71 15.74 2.71] Press RETURN Eigenvalues of T? [ 1. +0.j -0.09+0.73j -0.09-0.73j -0.53+0.j 0. +0.j -0. +0.j 0. +0.j ] Press RETURN A3: 7 nodes, with a source, a sink, and an island Use Google smoothing option. Small probability that anyone can pop over to anywhere. pagerank_test(): The digraph adjacency matrix A. [[0 1 0 0 0 1 0] [0 0 1 0 0 0 0] [1 0 0 1 0 0 0] [1 0 0 0 0 0 0] [0 0 1 0 0 0 0] [0 0 0 0 0 0 0] [0 0 0 0 0 0 0]] Press RETURN Plot the digraph A. Press RETURN Define transition matrix T: [[0.02 0.02 0.45 0.87 0.02 0.14 0.14] [0.45 0.02 0.02 0.02 0.02 0.14 0.14] [0.02 0.87 0.02 0.02 0.87 0.14 0.14] [0.02 0.02 0.45 0.02 0.02 0.14 0.14] [0.02 0.02 0.02 0.02 0.02 0.14 0.14] [0.45 0.02 0.02 0.02 0.02 0.14 0.14] [0.02 0.02 0.02 0.02 0.02 0.14 0.14]] Press RETURN Watch 100 people move around digraph, x <- T * x: 0 [100. 0. 0. 0. 0. 0. 0.] 1 [ 2.14 44.64 2.14 2.14 2.14 44.64 2.14] 2 [10.56 8.73 47.59 8.73 7.82 8.73 7.82] 3 [31.8 8.64 18.23 24.38 4.15 8.64 4.15] 4 [32.17 17.21 14.57 11.44 3.7 17.21 3.7 ] 5 [20.6 18.35 22.46 10.87 4.68 18.35 4.68] 6 [23.73 13.7 24.52 14.48 4.94 13.7 4.94] 7 [27.14 14.49 20.25 14.83 4.41 14.49 4.41] 8 [25.64 15.97 20.5 13.04 4.44 15.97 4.44] 9 [24.42 15.52 21.97 13.33 4.62 15.52 4.62] 10 [25.26 14.97 21.71 13.92 4.59 14.97 4.59] 11 [25.58 15.25 21.14 13.74 4.52 15.25 4.52] 12 [25.21 15.41 21.35 13.53 4.54 15.41 4.54] 13 [25.14 15.28 21.53 13.64 4.57 15.28 4.57] 14 [25.3 15.24 21.42 13.7 4.55 15.24 4.55] Press RETURN Eigenvalues of T? [ 1. +0.j -0.08+0.62j -0.08-0.62j -0.45+0.j 0. +0.j -0. +0.j -0. +0.j ] Press RETURN pagerank_test(): Normal end of execution.