Phase Lengths in the Cylic Cellular Automaton Kiran Tomlinson - - PowerPoint PPT Presentation

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Phase Lengths in the Cylic Cellular Automaton Kiran Tomlinson - - PowerPoint PPT Presentation

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Introduction Methods Results Discussion Phase Lengths in the Cylic Cellular Automaton Kiran Tomlinson Department of Computer Science, Carleton College CSC 2019 Introduction Methods Results Discussion Outline 1 Cyclic cellular automaton

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Introduction Methods Results Discussion

Phase Lengths in the Cylic Cellular Automaton

Kiran Tomlinson

Department of Computer Science, Carleton College

CSC 2019

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Introduction Methods Results Discussion

Outline

1 Cyclic cellular automaton (CCA) 2 Phases in the CCA 3 Measuring phase lengths 4 Experimental design 5 Results

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Introduction Methods Results Discussion

Spiral Wave Excitable Media

Figure from A. T. Winfree and S. H. Strogatz, “Organiz- ing centres for three-dimensional chemical waves,” Nature,

  • vol. 311, pp. 611–615, 1984.
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Introduction Methods Results Discussion

Cyclic Cellular Automaton (CCA)

(Fisch, Gravner, & Griffeath, 1991)

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Introduction Methods Results Discussion

Cyclic Cellular Automaton (CCA)

(Fisch, Gravner, & Griffeath, 1991)

k = 9 4 8 3 7 4 5 1 6 1 7 4 1 7 7 5 3 1 1 8 7 6

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Introduction Methods Results Discussion

Cyclic Cellular Automaton (CCA)

(Fisch, Gravner, & Griffeath, 1991)

4 8 3 7 4 5 1 6 1 7 4 1 7 7 5 3 1 1 8 7 6 4 8 3 7 4 5 1 6 1 7 4 1 7 7 5 3 1 1 8 7 6

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Introduction Methods Results Discussion

Cyclic Cellular Automaton (CCA)

(Fisch, Gravner, & Griffeath, 1991)

t → t + 1 4 8 3 7 4 5 1 6 1 7 4 1 7 7 5 3 1 1 8 7 6 4 8 3 7 4 5 1 6 1 7 4 1 7 7 5 3 1 1 8 7 6

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Introduction Methods Results Discussion

Cyclic Cellular Automaton (CCA)

(Fisch, Gravner, & Griffeath, 1991)

t → t + 1 4 8 3 7 4 5 1 6 1 7 4 1 7 7 5 3 1 1 8 7 6 4 8 3 7 4 5 1 6 1 7 4 1 7 7 5 3 1 1 8 7 6

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Introduction Methods Results Discussion

Cyclic Cellular Automaton (CCA)

(Fisch, Gravner, & Griffeath, 1991)

t → t + 1 4 8 3 7 4 5 1 6 1 7 4 1 7 7 5 3 1 1 8 7 6 4 8 3 7 4 5 1 6 1 7 4 1 7 7 5 3 1 1 8 7 6 5

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Introduction Methods Results Discussion

Cyclic Cellular Automaton (CCA)

Moore neighborhood 4 8 3 7 4 5 1 6 1 7 4 1 7 7 5 3 1 1 8 7 6 4 8 3 7 4 5 1 6 1 7 4 1 7 7 5 3 1 1 8 7 6 von Neumann neighborhood 4 8 3 7 4 5 1 6 1 7 4 1 7 7 5 3 1 1 8 7 6 4 8 3 7 4 5 1 6 1 7 4 1 7 7 5 3 1 1 8 7 6

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Introduction Methods Results Discussion

Formal Definition

Notation ζt(x) ∈ {0, 1, . . . , k − 1} state of cell x at time t N(x) neighbors of cell x N +

t (x)

promoters of cell x at time t Update Rule N +

t (x) = {y ∈ N(x) | (ζt(x) + 1) mod k = ζt(y)}

ζt+1(x) =

  • (ζt(x) + 1) mod k

if |N +

t (x)| ≥ 1

ζt(x)

  • therwise
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Introduction Methods Results Discussion

CCA Phases

(Fisch, Gravner, & Griffeath, 1991)

Debris Droplet Defect Demon

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Introduction Methods Results Discussion

Prior CCA Work

1 Classifying behavior in parameter space

(Fisch, Gravner, & Griffeath, 1991), (Durrett & Griffeath, 1993), (Hawick, 2013)

2 Effects of neighborhood on spiral shape

(Reiter, 2010)

3 Quantifying self-organization

(Shalizi & Shalizi, 2003)

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Introduction Methods Results Discussion

Prior CCA Work

1 Classifying behavior in parameter space

(Fisch, Gravner, & Griffeath, 1991), (Durrett & Griffeath, 1993), (Hawick, 2013)

2 Effects of neighborhood on spiral shape

(Reiter, 2010)

3 Quantifying self-organization

(Shalizi & Shalizi, 2003)

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Introduction Methods Results Discussion

Prior CCA Work

1 Classifying behavior in parameter space

(Fisch, Gravner, & Griffeath, 1991), (Durrett & Griffeath, 1993), (Hawick, 2013)

2 Effects of neighborhood on spiral shape

(Reiter, 2010)

3 Quantifying self-organization

(Shalizi & Shalizi, 2003)

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Introduction Methods Results Discussion Figure from R. Fisch, J. Gravner, and D. Griffeath, “Cyclic cellular automata in two dimensions,” in Spatial Stochastic

  • Processes. Springer, 1991, pp. 71–185.

How does the number of cell types affect phase lengths?

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Introduction Methods Results Discussion

Identifying Phase Transitions

∆(t) =

  • x

(ζt(x) − ζt−1(x) mod k)

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Introduction Methods Results Discussion

Identifying Phase Transitions

∆(t) =

  • x

(ζt(x) − ζt−1(x) mod k) = ⇒ transitions at local extrema of ∆′′(t)

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Introduction Methods Results Discussion

Noise Amplification

100 200 300 400 500 15000 30000 45000 60000

(t)

100 200 300 400 500

Step (t)

  • 100
  • 50

50 100

′′(t)

∆(t) curve (k = 13, von Neumann neighborhood)

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Introduction Methods Results Discussion

Savitzky-Golay Differentiation

(Savitzky & Golay, 1964) 1 Pick window width around point 2 Fit low-degree polynomial to data in window 3 Take derivative of fitted polynomial at point

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Introduction Methods Results Discussion

Savitzky-Golay Differentiation

(Savitzky & Golay, 1964) 1 Pick window width around point 2 Fit low-degree polynomial to data in window 3 Take derivative of fitted polynomial at point

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Introduction Methods Results Discussion

Savitzky-Golay Differentiation

(Savitzky & Golay, 1964) 1 Pick window width around point 2 Fit low-degree polynomial to data in window 3 Take derivative of fitted polynomial at point

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Introduction Methods Results Discussion

Identifying Phase Transitions

100 200 300 400 500 15000 30000 45000 60000

(t)

100 200 300 400 500

Step (t)

  • 100
  • 50

50 100

′′(t)

∆(t) curve (k = 13, von Neumann neighborhood)

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Introduction Methods Results Discussion

Identifying Phase Transitions

100 200 300 400 500 15000 30000 45000 60000

(t)

100 200 300 400 500

Step (t)

  • 100
  • 50

50 100

′′(t)

∆(t) curve (k = 13, von Neumann neighborhood)

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Introduction Methods Results Discussion

Simulation Procedure

1024 trials of 500 steps for each setting 3 grid sizes 2 neighborhoods k between 7 and 20

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Introduction Methods Results Discussion

Simulation Procedure

256

512

1024

1024 trials of 500 steps for each setting 3 grid sizes 2 neighborhoods k between 7 and 20

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Introduction Methods Results Discussion

Simulation Procedure

256

512

1024

Moore von Neumann 1024 trials of 500 steps for each setting 3 grid sizes 2 neighborhoods k between 7 and 20

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Introduction Methods Results Discussion

Simulation Procedure

256

512

1024

Moore von Neumann 1024 trials of 500 steps for each setting 3 grid sizes 2 neighborhoods k between 7 and 20

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Introduction Methods Results Discussion

Phase Length Dependence on k

For each parameter setting:

1 Compute phase lengths in each trial 2 Average over trials

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Introduction Methods Results Discussion

Phase Length Dependence on k

For each parameter setting:

1 Compute phase lengths in each trial 2 Average over trials

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Introduction Methods Results Discussion

Phase Length Dependence on k

For each parameter setting:

1 Compute phase lengths in each trial 2 Average over trials

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Introduction Methods Results Discussion

Phase Length Dependence on k

For each parameter setting:

1 Compute phase lengths in each trial 2 Average over trials 2.0 2.2 2.4 2.6 log k 1 2 3 4 5 log Phase Length

Debris

Grid size 256x256 512x512 1024x1024 2.0 2.2 2.4 2.6 log k

Droplet

2.0 2.2 2.4 2.6 log k

Defect

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Introduction Methods Results Discussion

Phase Length Dependence on k

For each parameter setting:

1 Compute phase lengths in each trial 2 Average over trials 2.0 2.2 2.4 2.6 log k 1 2 3 4 5 log Phase Length

Debris

Grid size 256x256 512x512 1024x1024 2.0 2.2 2.4 2.6 log k

Droplet

2.0 2.2 2.4 2.6 log k

Defect

log L = a + b log k L = akb

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Introduction Methods Results Discussion

Phase Length Dependence on k

For each parameter setting:

1 Compute phase lengths in each trial 2 Average over trials 2.0 2.2 2.4 2.6 log k 1 2 3 4 5 log Phase Length

Debris

Grid size 256x256 512x512 1024x1024 2.0 2.2 2.4 2.6 log k

Droplet

2.0 2.2 2.4 2.6 log k

Defect

log L = a + b log k L = akb

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Introduction Methods Results Discussion

Power Law Exponents and Coefficients

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Introduction Methods Results Discussion

Uncertainty Estimation

Random variance?

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Introduction Methods Results Discussion

Uncertainty Estimation

Random variance? = ⇒ bootstrapping

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Introduction Methods Results Discussion

Uncertainty Estimation

Random variance? = ⇒ bootstrapping Bias from Savitzky-Golay window widths?

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Introduction Methods Results Discussion

Uncertainty Estimation

Random variance? = ⇒ bootstrapping Bias from Savitzky-Golay window widths? = ⇒ perturb widths

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Introduction Methods Results Discussion

Uncertainty Estimation

Random variance? = ⇒ bootstrapping Bias from Savitzky-Golay window widths? = ⇒ perturb widths Conservative estimate: add confidence intervals

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Introduction Methods Results Discussion

Different Phase Length Sensitivities

Moore N Phase Exponent Debris 2.56 ± 0.01 Droplet 4.34 ± 0.03 Defect 2.81 ± 0.06 von Neumann N Phase Exponent Debris 2.56 ± 0.01 Droplet 4.81 ± 0.07 Defect 3.15 ± 0.06

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Introduction Methods Results Discussion

Different Phase Length Sensitivities

Moore N Phase Exponent Debris 2.56 ± 0.01 Droplet 4.34 ± 0.03 Defect 2.81 ± 0.06 von Neumann N Phase Exponent Debris 2.56 ± 0.01 Droplet 4.81 ± 0.07 Defect 3.15 ± 0.06 Smaller neighborhood = ⇒ more sensitive to k?

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Introduction Methods Results Discussion

Different Phase Length Sensitivities

Moore N Phase Exponent Debris 2.56 ± 0.01 Droplet 4.34 ± 0.03 Defect 2.81 ± 0.06 von Neumann N Phase Exponent Debris 2.56 ± 0.01 Droplet 4.81 ± 0.07 Defect 3.15 ± 0.06 Smaller neighborhood = ⇒ more sensitive to k? Droplet phase most sensitive

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Introduction Methods Results Discussion

Other Methods for Finding Transitions

1 2 3 4 3 4 5 6 5 4 5 6 7 7 8

Identify defects directly?

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Introduction Methods Results Discussion

Conclusions

2.0 2.2 2.4 2.6 log k 1 2 3 4 5 log Phase Length

Debris

Grid size 256x256 512x512 1024x1024 2.0 2.2 2.4 2.6 log k

Droplet

2.0 2.2 2.4 2.6 log k

Defect

1 CCA phase lengths depend on k via simple power laws 2 Independent of grid size 3 Exponent indicates sensitivity of phase to k

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Introduction Methods Results Discussion

Conclusions

2.0 2.2 2.4 2.6 log k 1 2 3 4 5 log Phase Length

Debris

Grid size 256x256 512x512 1024x1024 2.0 2.2 2.4 2.6 log k

Droplet

2.0 2.2 2.4 2.6 log k

Defect

1 CCA phase lengths depend on k via simple power laws 2 Independent of grid size 3 Exponent indicates sensitivity of phase to k

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Introduction Methods Results Discussion

Conclusions

2.0 2.2 2.4 2.6 log k 1 2 3 4 5 log Phase Length

Debris

Grid size 256x256 512x512 1024x1024 2.0 2.2 2.4 2.6 log k

Droplet

2.0 2.2 2.4 2.6 log k

Defect

1 CCA phase lengths depend on k via simple power laws 2 Independent of grid size 3 Exponent indicates sensitivity of phase to k

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Introduction Methods Results Discussion

Acknowledgments

Travel and conference funding from Carleton College Thanks to Frank McNally for feedback