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Cellular Automata

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Title: Cellular Automata, a Tool in Pure and Applied Mathematics Author: Irmacs Irmacs Last modified by: Vahid Created Date: 10/21/2010 5:56:55 PM – PowerPoint PPT presentation

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Title: Cellular Automata


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 Conways 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
  • Neighborhood
  • Black White neighbors (States)

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

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

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

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

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

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

t 1 t 2
17
1-D Cellular Automata
  • Neighborhood
  • Black White neighbors (States)
  • 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
Conways Game of Life
  • live
    dead

25
Conways Game of Life
  • live
    dead
  • 1. Any dead cell with exactly three live
    neighbors comes to life.

26
Conways 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
Conways 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
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29
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30
2-D Cellular Automata
31
2-D Cellular Automata
32
3-D Cellular Automata
33
3-D Cellular Automata
34
CA Model in Social Science
  • Criminal Activity
  • HIV Spread
  • Residential Migration
  • Crime and Liquor

35
2D-Cellular Automata
36
2D-Cellular Automata
37
Social Counter
38
Social Counter
  • N Neighbourhood of s
  • n Neighbours of s in N

N
s
39
Social Counter
  • N Neighbourhood of s
  • n Neighbours of s in N
  • an Social influence of n on s

N
s
40
Social Counter
  • N Neighbourhood of s
  • n Neighbours of s in N
  • an Social influence of n on s
  • Cs(t) Total influence on s at time t?

N
s
41
Social Counter
  • N Neighbourhood of s
  • n Neighbours of s in N
  • an Social influence of n on s
  • Cs(t) Total influence on s at time t?
  • Cs(t) Cs(t-1) Sn an

N
s
42
Social Counter
Environmental influence
  • N Neighbourhood of s
  • n Neighbours of s in N
  • an Social influence of n on s
  • Cs(t) Total influence on s at time t?
  • Cs(t) Cs(t-1) Sn an

N
s
43
Social Counter
Environmental influence
  • N Neighbourhood of s
  • n Neighbours of s in N
  • an Social influence of n on s
  • Cs(t) Total influence on s at time t?
  • Cs(t) Cs(t-1) Sn an
  • ß Environmental influence on s
  • Cs(t) Cs(t-1) Sn an ß

N
s
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.
45
Individuals (states)
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.

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.

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.

51
0 Stayers
3 HRDU
2 LRDU
1 Susceptible
4 Incapacitation
52
0 Stayers
3 HRDU
2 LRDU
1 Susceptible
P42
P43
4 Incapacitation
P41
P34
53
0 Stayers
3 HRDU
2 LRDU
1 Susceptible
4 Incapacitation
54
0 Stayers
3 HRDU
2 LRDU
1 Susceptible
4 Incapacitation
55
0 Stayers
3 HRDU
2 LRDU
1 Susceptible
4 Incapacitation
56
0 Stayers
3 HRDU
2 LRDU
1 Susceptible
4 Incapacitation
57
0 Stayers
3 HRDU
2 LRDU
1 Susceptible
4 Incapacitation
58
0 Stayers
3 HRDU
2 LRDU
1 Susceptible
4 Incapacitation
59
0 Stayers
3 HRDU
2 LRDU
1 Susceptible
4 Incapacitation
60
0 Stayers
v01
v03
v13
v02
v12
v23
3 HRDU
2 LRDU
1 Susceptible
v32
v21
P42
v32
P43
4 Incapacitation
P41
P34
61
Social Counters
  • C1(t) C1(t - 1) R0?01 R2?21 R3?31
  • C2(t) C2(t - 1) R0?02 R1 ?12 R3?32
  • C3(t) C3(t - 1) R0?03 R1 ?13 R2?23
  • Ri is the number cells of type i 0,,3 in a
    neighbourhood

62
0 Stayers
a
a
a
a
a
a
3 HRDU
2 LRDU
1 Susceptible


P42

P43
4 Incapacitation
P41
P34
63
Rules
  • Suppose 0 a, ß 1
  • Susceptible If C1(t) -1 then becomes LRDU.
  • LRDU
  • a). If C2(t) 1 then becomes Susceptible.
  • b). If C2(t) -1 then becomes HRDU
  • HRDU
  • a). If C3(t) 1 then becomes LRDU.
  • b). Moves to Incapacitation with probability
    P34.
  • Incapacitation Becomes a Susceptible, LRDU and
    HRDU with probabilities P41, P42 and P43,
    respectively.

64
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65
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66
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67
Modeling HIV Spread through Sexual Contact Using
a Cellular Automaton
  • 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.

68
HIV Spread
1 HIV HR Uk
3 HIV- HR
0 HIV Kw
4 HIV- LR
2 HIV LR Uk
69
HIV Spread
1 HIV HR Uk
3 HIV- HR
P
0 HIV Kw
4 HIV- LR
2 HIV LR Uk
P
Transition via HIV infection
70
HIV Spread
1 HIV HR Uk
3 HIV- HR
P
Q
0 HIV Kw
Q
4 HIV- LR
2 HIV LR Uk
P
Transition via HIV infection Transition via
HIV test
71
HIV Spread
1 HIV HR Uk
3 HIV- HR
0 HIV Kw
4 HIV- LR
2 HIV LR Uk
72
HIV Spread
1 HIV HR Uk
3 HIV- HR
0 HIV Kw
4 HIV- LR
2 HIV LR Uk
Social interaction for non-risky behaviour
73
HIV Spread
1 HIV HR Uk
3 HIV- HR
0 HIV Kw
4 HIV- LR
2 HIV LR Uk
Social interaction for non-risky behaviour
Transition via social interaction
74
HIV Spread
1 HIV HR Uk
3 HIV- HR
0 HIV Kw
4 HIV- LR
2 HIV LR Uk
Social interaction for risky behaviour Social
interaction for non-risky behaviour
Transition via social interaction
75
HIV Spread
1 HIV HR Uk
3 HIV- HR
0 HIV Kw
4 HIV- LR
2 HIV LR Uk
Social interaction for risky behaviour Social
interaction for non-risky behaviour
Transition via social interaction
76
HIV Spread
2 HIV HR Uk
3 HIV- HR
P
Q
0 HIV Kw
Q
4 HIV- LR
1 HIV LR Uk
P
Transition via social interaction Transition via
HIV infection Transition via HIV test
Social interaction for risky behaviour Social
interaction for non-risky behaviour
77
HIV Spread
V23
2 HIV HR Uk
3 HIV- HR
P
V03
V02
V32
Q
V31
V24
0 HIV Kw
V34
V43
V12
V21
V13
V42
V41
Q
4 HIV- LR
1 HIV LR Uk
V01
P
V04
V14
78
HIV Spread
V23
2 HIV HR Uk
3 HIV- HR
P
V03
V02
V32
Q
V31
V24
0 HIV Kw
V34
V43
V12
V21
V13
V42
V41
Q
4 HIV- LR
1 HIV LR Uk
V01
P
V04
V14
Ci(t) Ci(t-1) Sj vji for i 1,,4 j
0,,4
79
Randomized CA
80
Randomized CA
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.
82
Moore Neighvorhood
83
Assumptions
  • Household parameters Hij(t) s(t), Tij(t), T
  • s(t) The social structure at time t (S- s(t)
    S)
  • Tij(t) The length of stay in (i, j) at time t.
  • T Time to settle in to neighbourhood
  • Location parameters of (i, j)
  • Cij Maximum Capacity
  • Cij(t) Capacity at time t
  • Vij(t) Social value at time t (average social
    structure of neighbours of (i, j))

84
Rules
Update for s(t)      s(t) min s(t - 1)
e, S if s(t - 1) e gt 0 s(t) max
s(t - 1) e, S- if s(t - 1) e 0
  e is a randomly determined value with a normal
distribution centred on zero
85
Rules
Moving from (i, j)     if Tij (t) gt T then it
moves with the probability
P(t) Vij(t) - s(t) / (S -
S-)   Moving to (i, j)   if Cij gt Cij(t) and
s(t) Vij(t)
86
High versus Low Neighborhood Influence
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.
88
Crime and Liquor
89
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90
Level of Analysis
91
Level of Analysis
92
Level of Analysis
93
States
1 Low-Risk
3 High-Risk
2 Medium-Risk
94
States
1 Low-Risk
3 High-Risk
2 Medium-Risk
SL
95
States
1 Low-Risk
3 High-Risk
2 Medium-Risk
SM
96
States
1 Low-Risk
3 High-Risk
2 Medium-Risk
SH
97
Impact of Social Influence
98
Impact of Social Influence
99
Impact of Social Influence
100
Impact of Social Influence
101
Impact of Social Influence
102
Impact of Social Influence
103
Impact of Social Influence
104
Impact of Social Influence
105
Impact of Social Influence
106
Assumptions
  • nlij Number of licences
  • Pij(t) Risky population at time t
  • Pij(t) Pij (t-1) ? SnPn (t-1)
  • for Sn in SL, SM, SH

n in Nij
107
Crime and Liquor
n in Nij
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