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Cellular Automata

Cellular Automata. MATH 800 Fall 2011. “ Cellular Automata ”. 588,000 results in 94,600 results in 61,500 results in. Applications. Physics Chemistry Biology Mathematics Social Science Health Science Criminology. Cellular Automata - History.

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Cellular Automata

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  1. Cellular Automata MATH 800 Fall 2011

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

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

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

  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)

  6. Cellular Automata • 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.

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

  8. 1-D Cellular Automata

  9. 1-D Cellular Automata • Neighborhood

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

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

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

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

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

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

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

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

  18. 1-D Cellular Automata

  19. 1-D Cellular Automata

  20. 1-D Cellular Automata

  21. 1-D Cellular Automata

  22. 2-D Cellular Automata

  23. 2-D Cellular Automata Moore von Neumann Hexagonal

  24. Conway’s Game of Life live dead

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

  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.

  27. 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. • 3. Any live cell with two or three live neighbors lives, unchanged, to the next generation.

  28. 2-D Cellular Automata

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

  30. 3-D Cellular Automata

  31. 3-D Cellular Automata

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

  33. 2D-Cellular Automata

  34. 2D-Cellular Automata

  35. Social Counter

  36. Social Counter • N: Neighbourhood of s • n: Neighbours of s in N N s

  37. Social Counter • N: Neighbourhood of s • n: Neighbours of s in N • αn: Social influence of non s N s

  38. Social Counter • N: Neighbourhood of s • n: Neighbours of s in N • αn: Social influence of non s • Cs(t): Total influence on s at time t? N s

  39. Social Counter • N: Neighbourhood of s • n: Neighbours of s in N • αn: Social influence of non s • Cs(t): Total influence on s at time t? Cs(t) = Cs(t-1) + Σn αn N s

  40. Social Counter Environmental influence • N: Neighbourhood of s • n: Neighbours of s in N • αn: Social influence of non s • Cs(t): Total influence on s at time t? Cs(t) = Cs(t-1) + Σn αn N s

  41. Social Counter Environmental influence • N: Neighbourhood of s • n: Neighbours of s in N • αn: Social influence of non s • Cs(t): Total influence on s at time t? Cs(t) = Cs(t-1) + Σn αn • β: Environmental influence on s Cs(t) = Cs(t-1) + Σn αn + β N s

  42. 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.

  43. Individuals (states)

  44. 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.

  45. 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. 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. 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. 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.

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