1. Cellular Automata MATH 800 Fall 2011 1

2. “Cellular Automata” 588,000 results in 94,600 results in 61,500 results in 2

3. Applications Physics Chemistry Biology Mathematics Social Science Health Science Criminology 3

4. Cellular Automata - History von Neumann and Ulam (1940s) John Conway ’ s Game of Life (1970) Stephen Wolfram - Mathematica (1983) John von Neumann (1903- 1957) Stanislaw Ulam (1909 - 1984) John Conway Stephen Wolfram 4

5. CA Model in Social Science T. Schelling, Models of segregation (1969) J.M. Sakoda, The checkerboard model of social interaction (1971) P.S. Albin, The Analysis of Complex Socioeconomic Systems (1975) 5

6. A mathematical model of spatial interactions, in which cells on an array are assigned a particular state, which then changes stepwise according to specific rules conditioned on the states of neighboring cells. Cellular Automata 6

7. Discrete Dynamical System – Space and time Dimension – 1-D, 2-D, … States – Neighborhood and neighbors Rules 7

8. 1-D Cellular Automata 8

9. Neighborhood 9

10. 1-D Cellular Automata Black & White neighbors (States) Neighborhood 10

11. 1-D Cellular Automata Black & White neighbors (States) Neighborhood Rules 11

12. 1-D Cellular Automata Black & White neighbors (States) Neighborhood Rules Example t = 1 12

13. 1-D Cellular Automata Black & White neighbors (States) Neighborhood Rules Example t = 1 t = 2 13

14. 1-D Cellular Automata Black & White neighbors (States) Neighborhood Rules Example t = 1 t = 2 14

15. 1-D Cellular Automata Black & White neighbors (States) Neighborhood Rules Example t = 1 t = 2 15

16. 1-D Cellular Automata Black & White neighbors (States) Neighborhood Rules Example t = 1 t = 2 16

17. 1-D Cellular Automata Black & White neighbors (States) Neighborhood Rules Example t = 1 t = 2 17

18. 1-D Cellular Automata 18

19. 1-D Cellular Automata 19

20. 1-D Cellular Automata 20

21. 1-D Cellular Automata 21

22. 2-D Cellular Automata 22

23. 2-D Cellular Automata von Neumann Moore Hexagonal 23

24. Conway’s Game of Life live dead 24

25. Conway’s Game of Life live dead 1. Any dead cell with exactly three live neighbors comes to life. 25

26. Conway’s Game of Life live dead 1. Any dead cell with exactly three live neighbors comes to life. 2. Any live cell with fewer than two live neighbours (loneliness), or more than three live neighbours (overcrowding) dies. 26

27. Conway’s Game of Life live dead 1. Any dead cell with exactly three live neighbors comes to life. 3. Any live cell with two or three live neighbors lives, unchanged, to the next generation. 2. Any live cell with fewer than two live neighbours (loneliness), or more than three live neighbours (overcrowding) dies. 27

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30. 2-D Cellular Automata 30

31. time x time y time z 2-D Cellular Automata 31

32. 3-D Cellular Automata 32

33. 3-D Cellular Automata 33

34. CA Model in Social Science Criminal Activity HIV Spread Residential Migration Crime and Liquor 34

35. 2D-Cellular Automata 35

36. 2D-Cellular Automata 36

37. Social Counter 37

38. Social Counter N: Neighbourhood of s n: Neighbours of s in N s s N 38

39. Social Counter N: Neighbourhood of s n: Neighbours of s in N α n : Social influence of n on s s s N 39

40. Social Counter N: Neighbourhood of s n: Neighbours of s in N α n : Social influence of n on s C s (t): Total influence on s at time t? s s N 40

41. Social Counter N: Neighbourhood of s n: Neighbours of s in N α n : Social influence of n on s C s (t): Total influence on s at time t? C s (t) = C s (t-1) + Σ n α n s s N 41

42. Social Counter N: Neighbourhood of s n: Neighbours of s in N α n : Social influence of n on s C s (t): Total influence on s at time t? C s (t) = C s (t-1) + Σ n α n s s N Environmental influence 42

43. Social Counter N: Neighbourhood of s n: Neighbours of s in N α n : Social influence of n on s C s (t): Total influence on s at time t? C s (t) = C s (t-1) + Σ n α n β: Environmental influence on s C s (t) = C s (t-1) + Σ n α n + β s s N Environmental influence 43

44. The Social Impact in a High-Risk Community: A Cellular Automata Model V. Dabbaghian, V. Spicer, S.K. Singh, P. Borwein and P.L. Brantingham, The social impact in a high-risk community: a cellular automata model, Journal of Computational Science, 2 (2011) 238 – 246. 44

45. Individuals (states) 45

46. Individuals (states) Stayer: A person who does not commit crime or use drugs under any circumstances Susceptible: An individual who does not currently use drugs or commit crime, but may be incited to be a LRDU. LRDU: An individual that can become addicted to drug and become a HRDU. HRDU: An individual who is physiologically and psychologically addicted to hard drugs and his/her criminal behaviour is primarily motivated by drug acquisition. Incapacitation: Temporary removal of HRDU from the community because of arresting or possible rehabilitation. 46

47. Individuals (states) Stayer: A person who does not commit crime or use drugs under any circumstances Susceptible: An individual who does not currently use drugs or commit crime, but may be incited to be a LRDU. LRDU: An individual that can become addicted to drug and become a HRDU. HRDU: An individual who is physiologically and psychologically addicted to hard drugs and his/her criminal behaviour is primarily motivated by drug acquisition. Incapacitation: Temporary removal of HRDU from the community because of arresting or possible rehabilitation. 47

48. Individuals (states) Stayer: A person who does not commit crime or use drugs under any circumstances Susceptible: An individual who does not currently use drugs or commit crime, but may be incited to be a LRDU. LRDU: An individual that can become addicted to drug and become a HRDU. HRDU: An individual who is physiologically and psychologically addicted to hard drugs and his/her criminal behaviour is primarily motivated by drug acquisition. Incapacitation: Temporary removal of HRDU from the community because of arresting or possible rehabilitation. 48

49. Individuals (states) Stayer: A person who does not commit crime or use drugs under any circumstances Susceptible: An individual who does not currently use drugs or commit crime, but may be incited to be a LRDU. LRDU: An individual that can become addicted to drug and become a HRDU. HRDU: An individual who is physiologically and psychologically addicted to hard drugs and his/her criminal behaviour is primarily motivated by drug acquisition. Incapacitation: Temporary removal of HRDU from the community because of arresting or possible rehabilitation. 49

50. Individuals (states) Stayer: A person who does not commit crime or use drugs under any circumstances Susceptible: An individual who does not currently use drugs or commit crime, but may be incited to be a LRDU. LRDU: An individual that can become addicted to drug and become a HRDU. HRDU: An individual who is physiologically and psychologically addicted to hard drugs and his/her criminal behaviour is primarily motivated by drug acquisition. Incapacitation: Temporary removal of HRDU from the community because of arresting or possible rehabilitation. 50

51. 0 Stayers 3 HRDU 2 LRDU 1 Susceptible 4 Incapacitation 51

52. 0 Stayers 3 HRDU 2 LRDU 1 Susceptible 4 Incapacitation P 42 P 41 P 43 P 34 52

53. 0 Stayers 3 HRDU 2 LRDU 1 Susceptible 4 Incapacitation 53

54. 0 Stayers 3 HRDU 2 LRDU 1 Susceptible 4 Incapacitation 54

55. 0 Stayers 3 HRDU 2 LRDU 1 Susceptible 4 Incapacitation 55

56. 0 Stayers 3 HRDU 2 LRDU 1 Susceptible 4 Incapacitation 56

57. 0 Stayers 3 HRDU 2 LRDU 1 Susceptible 4 Incapacitation 57

58. 0 Stayers 3 HRDU 2 LRDU 1 Susceptible 4 Incapacitation 58

59. 0 Stayers 3 HRDU 2 LRDU 1 Susceptible 4 Incapacitation 59

60. 0 Stayers 3 HRDU 2 LRDU 1 Susceptible 4 Incapacitation v 01 v 03 v 02 v 13 v 32 v 21 v 12 v 23 v 32 P 42 P 41 P 43 P 34 60

61. Social Counters C 1 (t) = C 1 (t - 1) + R 0 01 + R 2 21 + R 3 31 C 2 (t) = C 2 (t - 1) + R 0 02 + R 1 12 + R 3 32 C 3 (t) = C 3 (t - 1) + R 0 03 + R 1 13 + R 2 23 R i is the number cells of type i = 0,…,3 in a neighbourhood 61

62. 0 Stayers 3 HRDU 2 LRDU 1 Susceptible 4 Incapacitation α α α α -β αα P 42 P 41 P 43 P 34 62

63. Rules Suppose 0 ≤ α, β ≤ 1 Susceptible: If C 1 (t) ≤ -1 then becomes LRDU. LRDU: – a). If C 2 (t) ≥ 1 then becomes Susceptible. – b). If C 2 (t) ≤ -1 then becomes HRDU HRDU: – a). If C 3 (t) ≥ 1 then becomes LRDU. – b). Moves to Incapacitation with probability P 34. Incapacitation: Becomes a Susceptible, LRDU and HRDU with probabilities P 41, P 42 and P 43, respectively. 63

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67. Modeling HIV Spread through Sexual Contact Using a Cellular Automaton A.Alimadad, V. Dabbaghian, S.K. Singh and H.H. Tsang, Modeling HIV Spread through Sexual Contact Using a Cellular Automaton, IEEE Congress on Evolutionary Computation, (2011), 2345 - 2350. 67

68. 4 HIV - LR 4 HIV - LR 0 HIV + Kw 0 HIV + Kw 1 HIV + HR Uk 1 HIV + HR Uk 2 HIV + LR Uk 2 HIV + LR Uk 3 HIV - HR 3 HIV - HR HIV Spread 68

69. 4 HIV - LR 4 HIV - LR 0 HIV + Kw 0 HIV + Kw 1 HIV + HR Uk 1 HIV + HR Uk 2 HIV + LR Uk 2 HIV + LR Uk 3 HIV - HR 3 HIV - HR Transition via HIV infection HIV Spread P P 69

70. 4 HIV - LR 4 HIV - LR 0 HIV + Kw 0 HIV + Kw 1 HIV + HR Uk 1 HIV + HR Uk 2 HIV + LR Uk 2 HIV + LR Uk 3 HIV - HR 3 HIV - HR Transition via HIV infection Transition via HIV test HIV Spread P P Q Q 70

71. 4 HIV - LR 4 HIV - LR 0 HIV + Kw 0 HIV + Kw 1 HIV + HR Uk 1 HIV + HR Uk 2 HIV + LR Uk 2 HIV + LR Uk 3 HIV - HR 3 HIV - HR HIV Spread 71

72. 4 HIV - LR 4 HIV - LR 0 HIV + Kw 0 HIV + Kw 1 HIV + HR Uk 1 HIV + HR Uk 2 HIV + LR Uk 2 HIV + LR Uk 3 HIV - HR 3 HIV - HR Social interaction for non-risky behaviour HIV Spread 72

73. 4 HIV - LR 4 HIV - LR 0 HIV + Kw 0 HIV + Kw 1 HIV + HR Uk 1 HIV + HR Uk 2 HIV + LR Uk 2 HIV + LR Uk 3 HIV - HR 3 HIV - HR Social interaction for non-risky behaviour HIV Spread Transition via social interaction 73

74. 4 HIV - LR 4 HIV - LR 0 HIV + Kw 0 HIV + Kw 1 HIV + HR Uk 1 HIV + HR Uk 2 HIV + LR Uk 2 HIV + LR Uk 3 HIV - HR 3 HIV - HR Social interaction for risky behaviour Social interaction for non-risky behaviour HIV Spread Transition via social interaction 74

75. 4 HIV - LR 4 HIV - LR 0 HIV + Kw 0 HIV + Kw 1 HIV + HR Uk 1 HIV + HR Uk 2 HIV + LR Uk 2 HIV + LR Uk 3 HIV - HR 3 HIV - HR Social interaction for risky behaviour Social interaction for non-risky behaviour HIV Spread Transition via social interaction 75

76. 4 HIV - LR 4 HIV - LR 0 HIV + Kw 0 HIV + Kw 2 HIV + HR Uk 2 HIV + HR Uk 1 HIV + LR Uk 1 HIV + LR Uk 3 HIV - HR 3 HIV - HR Transition via social interaction Transition via HIV infection Transition via HIV test Social interaction for risky behaviour Social interaction for non-risky behaviour HIV Spread P P Q Q 76

77. 4 HIV - LR 4 HIV - LR 0 HIV + Kw 0 HIV + Kw 2 HIV + HR Uk 2 HIV + HR Uk 1 HIV + LR Uk 1 HIV + LR Uk 3 HIV - HR 3 HIV - HR V 03 V 04 V 02 V 01 V 13 V 31 V 24 V 42 V 32 V 23 V 41 V 14 V 34 V 43 V 12 V 21 HIV Spread P P Q Q 77

78. 4 HIV - LR 4 HIV - LR 0 HIV + Kw 0 HIV + Kw 2 HIV + HR Uk 2 HIV + HR Uk 1 HIV + LR Uk 1 HIV + LR Uk 3 HIV - HR 3 HIV - HR V 03 V 04 V 02 V 01 V 13 V 31 V 24 V 42 V 32 V 23 V 41 V 14 V 34 V 43 V 12 V 21 HIV Spread C i (t) = C i (t-1) + Σ j v ji for i = 1,…,4 & j =0,…,4 P P Q Q 78

79. Randomized CA 79

80. Randomized CA 80

81. A cellular automata model on residential migration in response to neighborhood social dynamics V. Dabbaghian, P. Jackson, V. Spicer and K. Wuschke. A cellular automata model on residential migration in response to neighborhood social dynamics. Math. Comput. Modelling, 52 (2010), 1752 - 1762. 81

82. Moore Neighvorhood 82

83. Assumptions Household parameters H ij (t) = [s(t), T ij (t), T] o s(t): The social structure at time t (S - ≤ s(t) ≤ S + ) o T ij (t): The length of stay in (i, j) at time t. o T: Time to settle in to neighbourhood Location parameters of (i, j) o C ij : Maximum Capacity o C ij (t): Capacity at time t o V ij (t): Social value at time t (average social structure of neighbours of (i, j)) 83

84. Rules Update for s(t) s(t) = min {s(t − 1) + ε, S + } if s(t − 1) + ε > 0 s(t) = max {s(t − 1) + ε, S - } if s(t − 1) + ε ≤ 0 ε is a randomly determined value with a normal distribution centred on zero 84

85. Rules Moving from (i, j) if T ij (t) > T then it moves with the probability P(t) = |V ij (t) − s(t)| / (S + − S - ) Moving to (i, j) if C ij > C ij (t) and s(t) ≈ V ij (t) 85

86. High versus Low Neighborhood Influence 86

87. Bars on Blocks: A Cellular Automata Model of Crime and Liquor Licensed Establishment Density V. Spicer, J. Ginther, H. Seifi, A. A. Reid and V. Dabbaghian. Bars on blocks: a cellular automata model of crime and liquor licensed establishment density, submitted. 87

88. Crime and Liquor 88

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90. Level of Analysis 90

91. Level of Analysis 91

92. Level of Analysis 92

93. States 1 Low-Risk 3 High-Risk 2 Medium-Risk 93

94. 1 Low-Risk 3 High-Risk 2 Medium-Risk States SL 94

95. 1 Low-Risk 3 High-Risk 2 Medium-Risk States SM 95

96. 1 Low-Risk 3 High-Risk 2 Medium-Risk States SH 96

97. B Impact of Social Influence 97

98. B Impact of Social Influence 98

99. B Impact of Social Influence 99

100. B Impact of Social Influence 100

101. B Impact of Social Influence 101

102. B Impact of Social Influence 102

103. B Impact of Social Influence 103

104. B Impact of Social Influence 104

105. B Impact of Social Influence 105

106. nl ij : Number of licences P ij (t): Risky population at time t P ij (t) = P ij (t-1) + ∑ S n P n (t-1) for S n in {SL, SM, SH} Assumptions n in N ij 106

107. Crime and Liquor n in N ij 107