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

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

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.

Introduction Methods Results Discussion Cyclic Cellular Automaton (CCA) (Fisch, Gravner, & Griffeath, 1991)

Introduction Methods Results Discussion Cyclic Cellular Automaton (CCA) k = 9 7 6 7 1 6 3 1 1 3 7 8 5 4 5 8 4 0 7 7 0 0 4 1 0 1 (Fisch, Gravner, & Griffeath, 1991)

Introduction Methods Results Discussion Cyclic Cellular Automaton (CCA) 7 7 6 6 7 7 1 1 6 6 3 3 1 1 1 1 3 3 7 7 8 8 5 5 4 4 5 5 8 8 4 4 0 0 7 7 7 7 0 0 0 0 4 4 1 1 0 0 1 1 (Fisch, Gravner, & Griffeath, 1991)

Introduction Methods Results Discussion Cyclic Cellular Automaton (CCA) t → t + 1 7 7 6 6 7 7 1 1 6 6 3 3 1 1 1 1 3 3 7 7 8 8 5 5 4 4 5 5 8 8 4 4 0 0 7 7 7 7 0 0 0 0 4 4 1 1 0 0 1 1 (Fisch, Gravner, & Griffeath, 1991)

Introduction Methods Results Discussion Cyclic Cellular Automaton (CCA) t → t + 1 7 7 6 6 7 7 1 1 6 6 3 3 1 1 1 1 3 3 7 7 8 8 5 5 4 4 5 5 8 8 4 4 0 0 7 7 7 7 0 0 0 0 4 4 1 1 0 0 1 1 (Fisch, Gravner, & Griffeath, 1991)

Introduction Methods Results Discussion Cyclic Cellular Automaton (CCA) t → t + 1 7 7 6 6 7 7 1 1 6 6 3 3 1 1 1 1 3 3 7 7 8 8 5 5 4 5 4 5 5 8 8 4 4 0 0 7 7 7 7 0 0 0 0 4 4 1 1 0 0 1 1 (Fisch, Gravner, & Griffeath, 1991)

Introduction Methods Results Discussion Cyclic Cellular Automaton (CCA) Moore neighborhood von Neumann neighborhood 7 7 6 6 7 7 1 1 6 6 7 7 6 6 7 7 1 1 6 6 3 3 1 1 1 1 3 3 7 7 3 3 1 1 1 1 3 3 7 7 8 8 5 5 4 4 5 5 8 8 8 8 5 5 4 4 5 5 8 8 4 4 0 0 7 7 7 7 0 0 4 4 0 0 7 7 7 7 0 0 0 0 4 4 1 1 0 0 1 1 0 0 4 4 1 1 0 0 1 1

Introduction Methods Results Discussion Formal Definition Notation ζ t ( 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 ) | ( ζ t ( x ) + 1) mod k = ζ t ( y ) } � if |N + t ( x ) | ≥ 1 ( ζ t ( x ) + 1) mod k ζ t +1 ( x ) = ζ t ( x ) otherwise

Introduction Methods Results Discussion CCA Phases Droplet Debris Defect Demon (Fisch, Gravner, & Griffeath, 1991)

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)

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)

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)

Introduction Methods Results Discussion How does the number of cell types affect phase lengths? Figure from R. Fisch, J. Gravner, and D. Griffeath, “Cyclic cellular automata in two dimensions,” in Spatial Stochastic Processes . Springer, 1991, pp. 71–185.

Introduction Methods Results Discussion Identifying Phase Transitions � ( ζ t ( x ) − ζ t − 1 ( x ) mod k ) ∆( t ) = x

Introduction Methods Results Discussion Identifying Phase Transitions � ( ζ t ( x ) − ζ t − 1 ( x ) mod k ) ∆( t ) = x transitions at local extrema of ∆ ′′ ( t ) = ⇒

Introduction Methods Results Discussion Noise Amplification ∆( t ) curve ( k = 13, von Neumann neighborhood) 60000 45000 ( t ) 30000 15000 0 0 100 200 300 400 500 100 50 ′′ ( t ) 0 -50 -100 0 100 200 300 400 500 Step ( t )

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

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

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

Introduction Methods Results Discussion Identifying Phase Transitions ∆( t ) curve ( k = 13, von Neumann neighborhood) 60000 45000 ( t ) 30000 15000 0 0 100 200 300 400 500 100 50 ′′ ( t ) 0 -50 -100 0 100 200 300 400 500 Step ( t )

Introduction Methods Results Discussion Identifying Phase Transitions ∆( t ) curve ( k = 13, von Neumann neighborhood) 60000 45000 ( t ) 30000 15000 0 0 100 200 300 400 500 100 50 ′′ ( t ) 0 -50 -100 0 100 200 300 400 500 Step ( t )

Introduction Methods Results Discussion Simulation Procedure 1024 trials of 500 steps for each setting 3 grid sizes 2 neighborhoods k between 7 and 20

Introduction Methods Results Discussion Simulation Procedure 1024 512 256 1024 trials of 500 steps for each setting 3 grid sizes 2 neighborhoods k between 7 and 20

Introduction Methods Results Discussion Simulation Procedure Moore von Neumann 1024 512 256 1024 trials of 500 steps for each setting 3 grid sizes 2 neighborhoods k between 7 and 20

Introduction Methods Results Discussion Simulation Procedure Moore von Neumann 1024 512 256 1024 trials of 500 steps for each setting 3 grid sizes 2 neighborhoods k between 7 and 20

Introduction Methods Results Discussion Phase Length Dependence on k For each parameter setting: 1 Compute phase lengths in each trial 2 Average over trials

Introduction Methods Results Discussion Phase Length Dependence on k For each parameter setting: 1 Compute phase lengths in each trial 2 Average over trials

Introduction Methods Results Discussion Phase Length Dependence on k For each parameter setting: 1 Compute phase lengths in each trial 2 Average over trials

Introduction Methods Results Discussion Phase Length Dependence on k For each parameter setting: 1 Compute phase lengths in each trial 2 Average over trials Debris Droplet Defect Grid size 5 256x256 log Phase Length 512x512 4 1024x1024 3 2 1 2 . 0 2 . 2 2 . 4 2 . 6 2 . 0 2 . 2 2 . 4 2 . 6 2 . 0 2 . 2 2 . 4 2 . 6 log k log k log k

Introduction Methods Results Discussion Phase Length Dependence on k For each parameter setting: 1 Compute phase lengths in each trial 2 Average over trials Debris Droplet Defect Grid size 5 256x256 log Phase Length 512x512 4 1024x1024 3 2 1 2 . 0 2 . 2 2 . 4 2 . 6 2 . 0 2 . 2 2 . 4 2 . 6 2 . 0 2 . 2 2 . 4 2 . 6 log k log k log k log L = a + b log k L = ak b

Introduction Methods Results Discussion Phase Length Dependence on k For each parameter setting: 1 Compute phase lengths in each trial 2 Average over trials Debris Droplet Defect Grid size 5 256x256 log Phase Length 512x512 4 1024x1024 3 2 1 2 . 0 2 . 2 2 . 4 2 . 6 2 . 0 2 . 2 2 . 4 2 . 6 2 . 0 2 . 2 2 . 4 2 . 6 log k log k log k log L = a + b log k L = a k b

Introduction Methods Results Discussion Power Law Exponents and Coefficients

Introduction Methods Results Discussion Uncertainty Estimation Random variance?

Introduction Methods Results Discussion Uncertainty Estimation Random variance? = ⇒ bootstrapping

Introduction Methods Results Discussion Uncertainty Estimation Random variance? = ⇒ bootstrapping Bias from Savitzky-Golay window widths?

Introduction Methods Results Discussion Uncertainty Estimation Random variance? = ⇒ bootstrapping Bias from Savitzky-Golay window widths? = ⇒ perturb widths

Introduction Methods Results Discussion Uncertainty Estimation Random variance? = ⇒ bootstrapping Bias from Savitzky-Golay window widths? = ⇒ perturb widths Conservative estimate: add confidence intervals

Introduction Methods Results Discussion Different Phase Length Sensitivities Moore N von Neumann N Phase Exponent Phase Exponent 2 . 56 ± 0 . 01 2 . 56 ± 0 . 01 Debris Debris Droplet 4 . 34 ± 0 . 03 Droplet 4 . 81 ± 0 . 07 2 . 81 ± 0 . 06 3 . 15 ± 0 . 06 Defect Defect