Spreading Phenomena Spreading Phenomena and and Excitable Media Excitable Media Christian Jacob AI AI Department of Computer Science University of Calgary CPSC 565 — Winter 2003 Emergent Computing — CPSC 565 — Winter 2003 1 Christian Jacob, University of Calgary
Two-Dimensional Two-Dimensional Cellular Automata Cellular Automata Emergent Computing — CPSC 565 — Winter 2003 2 Christian Jacob, University of Calgary
Two-dimensional CA Neighbourhoods Two-dimensional CA Neighbourhoods von Neumann Moore Extended Moore r = 1 r = 1 r = 1, 2, 3 Formally, we can characterize the 2D Moore neighbourhood N ij ( r ) of radius r in a cell lattice L for a cell c ij by N ij ( r ) = {( k , l ) Œ L | | k - i | ≤ r and | l - j | ≤ r }. Emergent Computing — CPSC 565 — Winter 2003 3 Christian Jacob, University of Calgary
Totalistic Cellular Automata Totalistic Cellular Automata • Consider the following 1D CA rule: 111 110 101 100 011 010 001 000 0 1 1 0 1 0 0 0 • This automaton changes c i into a 1 only if the sum of the cells c i-1 , c i , and c i+1 is exactly 2: 1, if c i -1 ( t ) + c i ( t ) + c i +1 ( t ) = 2 { c i ( t+ 1) = 0, otherwise This is an example of a totalistic rule. Emergent Computing — CPSC 565 — Winter 2003 4 Christian Jacob, University of Calgary
Encoding of Totalistic Rules Encoding of Totalistic Rules • We first look at one-dimensional, binary CA rules ( k = 2) : c i ( t +1) = f ( ∑ j Œ N i(2) c i + j ( t ) ), where f : {0, 1, …, 2 r +1} Æ {0, 1}. • We can make a table of f in the form of a tuple ( f (0), f (1), f (2), …, f (2 r + 1) ). • We can encode this tuple by the following formula: C f = ∑ j=0, …, 2 r +1 f ( j ) · 2 j . Emergent Computing — CPSC 565 — Winter 2003 5 Christian Jacob, University of Calgary
Encoding of Totalistic Rules (2) Encoding of Totalistic Rules (2) • With this encoding all legal rule codes are even numbers. – All 32 rules for k = 2 and r = 2 are encoded by all even numbers between 0 and 62. Note : In the previous lecture we used a different notation, with V instead of k , and K = 2 r +1. – Example: mod-2 rule Sum: 0 1 2 3 4 5 f (Sum): 0 1 0 1 0 1 C f : 0 · 2 0 + 1 · 2 1 + 0 · 2 2 + 1 · 2 3 + 0 · 2 4 + 1 · 2 5 = 42 Emergent Computing — CPSC 565 — Winter 2003 6 Christian Jacob, University of Calgary
Encoding of Totalistic Rules (3) Encoding of Totalistic Rules (3) • This encoding for binary, totalistic CAs ( k = 2) C f = ∑ j=0, …, 2 r +1 f ( j ) · 2 j can be generalized to any number of k values per cell: C f = ∑ j=0, …, 2 r +1 f ( j ) · k j , where f maps to any of the k values for a neighbourhood of radius r : f : {0, 1, …, 2 r +1} Æ {0, 1, …, k }. Emergent Computing — CPSC 565 — Winter 2003 7 Christian Jacob, University of Calgary
Some Example 2D Cellular Automata: 1022 1022 Some Example 2D Cellular Automata: 5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic 0-39 20 30 40 Emergent Computing — CPSC 565 — Winter 2003 8 Christian Jacob, University of Calgary
Some Example 2D Cellular Automata: 510 510 Some Example 2D Cellular Automata: 5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic 0-39 20 30 40 Emergent Computing — CPSC 565 — Winter 2003 9 Christian Jacob, University of Calgary
Some Example 2D Cellular Automata: 374 374 Some Example 2D Cellular Automata: 5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic 0-39 20 30 40 Emergent Computing — CPSC 565 — Winter 2003 10 Christian Jacob, University of Calgary
Some Example 2D Cellular Automata: 614 614 Some Example 2D Cellular Automata: 5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic 0-39 20 30 40 Emergent Computing — CPSC 565 — Winter 2003 11 Christian Jacob, University of Calgary
Some Example 2D Cellular Automata: 174 174 Some Example 2D Cellular Automata: 5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic 0-39 20 30 40 Emergent Computing — CPSC 565 — Winter 2003 12 Christian Jacob, University of Calgary
Some Example 2D Cellular Automata: 494 494 Some Example 2D Cellular Automata: 5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic 0-39 20 30 40 Emergent Computing — CPSC 565 — Winter 2003 13 Christian Jacob, University of Calgary
Patterns from Seed Regions: 143954 143954 Patterns from Seed Regions: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Single-Cell Seed Random Disordered Seed Emergent Computing — CPSC 565 — Winter 2003 14 Christian Jacob, University of Calgary
Patterns from Seed Regions: 50224 50224 Patterns from Seed Regions: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Single-Cell Seed Random Disordered Seed Emergent Computing — CPSC 565 — Winter 2003 15 Christian Jacob, University of Calgary
Patterns from Seed Regions: 15822 15822 Patterns from Seed Regions: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Single-Cell Seed Random Disordered Seed Emergent Computing — CPSC 565 — Winter 2003 16 Christian Jacob, University of Calgary
Patterns from Seed Regions: 85507 85507 Patterns from Seed Regions: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Single-Cell Seed Random Disordered Seed Emergent Computing — CPSC 565 — Winter 2003 17 Christian Jacob, University of Calgary
Patterns from Seed Regions: 191044 191044 Patterns from Seed Regions: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Single-Cell Seed Random Disordered Seed Emergent Computing — CPSC 565 — Winter 2003 18 Christian Jacob, University of Calgary
Patterns from Seed Regions: 93737 93737 Patterns from Seed Regions: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Single-Cell Seed Random Disordered Seed Emergent Computing — CPSC 565 — Winter 2003 19 Christian Jacob, University of Calgary
Spreading Phenomena Spreading Phenomena Emergent Computing — CPSC 565 — Winter 2003 20 Christian Jacob, University of Calgary
Spreading Phenomena Spreading Phenomena • Spreading is the process in which an object extends itself over an increasingly larger area by incorporating regions adjacent to itself. • The spreading or kinetic growth (KG) models describe a wide variety of natural processes: – Tumor growth – Epidemic spread – Gelation – Rumor-mongering – Fluid flow through porous media – … Emergent Computing — CPSC 565 — Winter 2003 21 Christian Jacob, University of Calgary
Slow Diffusive Growth: 256746 256746 Slow Diffusive Growth: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Emergent Computing — CPSC 565 — Winter 2003 22 Christian Jacob, University of Calgary
Slow Diffusive Growth: 736 736 Slow Diffusive Growth: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Emergent Computing — CPSC 565 — Winter 2003 23 Christian Jacob, University of Calgary
Slow Diffusive Growth: 174826 174826 Slow Diffusive Growth: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Emergent Computing — CPSC 565 — Winter 2003 24 Christian Jacob, University of Calgary
Slow Diffusive Growth: 175850 175850 Slow Diffusive Growth: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Emergent Computing — CPSC 565 — Winter 2003 25 Christian Jacob, University of Calgary
Eden Model Eden Model • The original kinetic growth model was the Eden model , introduced by the biologist M. Eden in the 1960s. • The Eden model simulates accretive growth of a cell cluster (originally for simulating the spread of tumor cells) in a square lattice by sequentially adjoining randomly selected cells from the perimeter. • In more detail: – We start with a cluster list consisting of a seed site. – A site is randomly chosen from the perimeter list, consisting of the nearest sites adjacent to the seed site (v. Neumann neighbourhood). – The cluster and perimeter lists are updated. – Another site is randomly selected from the perimeter list, and so on. Emergent Computing — CPSC 565 — Winter 2003 26 Christian Jacob, University of Calgary
Variations of the Eden Model Variations of the Eden Model • We look at two variations of the Eden model: – The Invasion Percolation Model – The Single Percolation Cluster Model Emergent Computing — CPSC 565 — Winter 2003 27 Christian Jacob, University of Calgary
The Invasion Percolation Model The Invasion Percolation Model • The invasion percolation model describes the flow of fluid through porous media. • Invasion Percolation Algorithm: – Each site in the perimeter list has a random number associated with it. – The cluster spreads by incorporating the perimeter site with the lowest associated random number. • This model describes a process of spreading that “follows the path of least resistance.” Emergent Computing — CPSC 565 — Winter 2003 28 Christian Jacob, University of Calgary
Invasion Percolation Model: Spreading Invasion Percolation Model: Spreading Emergent Computing — CPSC 565 — Winter 2003 29 Christian Jacob, University of Calgary
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